The basic geometry of Witt vectors, I: The affine case

THE BASIC GEOMETR Y OF WITT VECTORS, I THE AFFINE CASE JAMES BORGER Abstract. W e give a concrete description of the category of ´ etale algebras o v er the ring o f Witt v e ctors of a giv en ﬁnite length with entries in an arbi trary ring. W e do this no t only for the classical p -t ypical and big Witt v e ctor functors but also for certain analogues o v er arbitrary lo cal and global ﬁelds. The basic theory of these generalized Witt ve ctors is developed from the p oint of view of commut ing F r ob enius li fts and their unive rsal prop erties, which is a new approac h even for the classi cal Witt vec tors. The l arger purp ose of this pap er is to pr o vide the aﬃne foundations for the algebraic geometry of generalized Witt schemes and arithmetic jet spaces. So the basics here ar e dev eloped somewhat f ully , with an ey e tow ard f uture applications. Introduction Witt vector functor s a re certain functors from the catego ry o f (commutativ e) rings to itself. The most common a re the p -typical Witt vector functors W , for each prime n um ber p . Given a ring A , one tr aditionally deﬁnes W ( A ) as a set to be A N and then g ives it the unique ring structure which is functorial in A and such that the set maps W ( A ) w − → A N ( x 0 , x 1 , . . . ) 7→ h x 0 , x p 0 + px 1 , x p 2 0 + px p 1 + p 2 x 2 , . . . i are ring ho momorphisms for all rings A , where the targ et has the ring structure with comp onent wise op era tions. F or example, w e hav e ( x 0 , x 1 , . . . ) + ( y 0 , y 1 , . . . ) = ( x 0 + y 0 , x 1 + y 1 − p − 1 X i =1 1 p  p i  x i 0 y p − i 0 , . . . ) ( x 0 , x 1 , . . . ) · ( y 0 , y 1 , . . . ) = ( x 0 y 0 , x p 0 y 1 + x 1 y p 0 + px 1 y 1 , . . . ) . Observe that the four p olyno mia ls in x 0 , y 0 , x 1 , y 1 display ed o n the rig h t-hand side hav e integer co e ﬃcie n ts, a s they must if they a re to deﬁne op eratio ns on W ( A ) for all rings A . Conv ersely , to prove that the des ired functor ial ring structure on W exists, it is e nough to prove tha t the p olyno mials sitting in the hig her c omp onents hav e integer co eﬃcients to o . This is Witt’s theore m. On the other hand, the p olynomials at the comp onent of index n dep end only on the v aria bles x 0 , y 0 , . . . , x n , y n . This is clear by induction. It follows that the quotient set A [0 ,n ] = { ( x 0 , . . . , x n ) } of W ( A ) = A N is a quo tient r ing, which we denote by W n ( A ). (It is traditionally deno ted W n +1 ( A ). The shift in indexing is preferable for reasons discussed in 2.5.) Date : October 22, 2018. 17:40. Mathematics Subje ct Classiﬁc ation (2010): 13F35. This work wa s partly supported by Disco ve ry Pr o j ect DP0773301, a grant from the Australian Researc h Council. 1 2 J. BOR GER In some cases, the ring s W ( A ) and W n ( A ) a re isomorphic to familiar r ings. F or example, W ( Z /p Z ) is is omorphic to the r ing Z p of p - a dic integers, and W n ( Z /p Z ) is isomorphic to Z / p n +1 Z . If p is inv ertible in A , then w is a bijection and so the Witt vector rings b ecome pro duct rings: W n ( A ) ∼ = A [0 ,n ] and W ( A ) ∼ = A N . But in most cases , W ( A ) is no t a familiar ring. While this tra ditional a pproach to Witt v ectors is adequa te for many purp os e s , it has tw o shortcoming s . T he ﬁrst is that it is not clear how we should think a b o ut the aﬃne scheme Sp e c W n ( A ) ge ometrically . Indeed, I am not aware of a truly geometric des cription of Sp ec W n ( A ) in any nontrivial c a se in the literature. If we wan t to fully incorp o rate Witt vectors into arithmetic a lgebraic geometry (and we do), it is impo rtant to have a thorough unders tanding of their geo metry . The main po int here a nd in the following pap er [4] is to set up a fr a mework for that. The geometry in this pap er is how ev er limited to the basic results in the aﬃne c a se needed for the genera l tr eatment in [4]. The second shortco ming of the traditional approa ch is that it do es no t explain what the deﬁning purp os e o f Witt vectors is. The answer, at least for this pa pe r, is that they co nt rol F robenius lifts—ring endomorphisms which reduce to the F rob e- nius map mo dulo p . Here we are follo wing Borger– Wieland [5], 12.3– 4, whic h in turn follow ed J oyal [18][19]. Motiv ated by this p ersp ective, we will de ﬁne Witt vec- tor functors relative to primes in any global or loc al ﬁeld. This genera lit y includes not only the p - typical functors ab ov e but also the so-c a lled big Witt vector functor and les s common v ar iants of the p -typical o nes due to Drinfeld a nd to Hazewinkel ([10], Pr op osition 1 .1; [16], (18.6.1 3)). I t also includes man y v ariants unstudied till now. W e will work with these generalized functors throughout the pap er. In fact, this will take no mo re eﬀort o nce we es tablish some basic reduction results. Let us now discuss the conten ts in more detail. Section 1 intro duces our gene r alized Witt vectors. Given a Dedekind domain R and a set E of maxima l ideals of R with ﬁnite residue ﬁelds, we will deﬁne a functor W R,E from the category Ring R of R -algebr as to itself: W R,E : Ri ng R → Ring R . (In fact, we will work with slightly more genera l R and E .) W e call W R,E the E - t ypical Witt vector functor. When R = Z and E consis ts of a single ma ximal ideal p Z , o ur functor will agree with the p -typical Witt vector functor ab ov e; when E consists o f all maximal ideals of Z , our functor will agre e with the big Witt vector functor. The deﬁnition of W R,E is in tw o steps. First w e deﬁne a functor W ﬂ R,E : Ri ng ﬂ R → Ring ﬂ R , where Ring ﬂ R is the full sub categor y of R ing R consisting of R - algebra s which ar e m -torsio n free for a ll ideals m ∈ E . W e call suc h algebra s E -ﬂat. Then w e deﬁne W R,E to b e a certain na tural extensio n of W ﬂ R,E to a ll of R ing R . Let N ( E ) denote the commutativ e monoid L E N , where N is { 0 , 1 , . . . } under addition. Given a n action of N ( E ) on an R -a lgebra B , let ψ m denote the endomor - phism of B g iven b y the m -th element o f the s ta ndard basis of N ( E ) . Let us say that s uch an action is a Λ R,E -structure if for each m ∈ E , the map ψ m reduces to the F ro be nius endomo rphism x 7→ x [ R : m ] on B / m B . Now, for any R -algebra A , the mono id N ( E ) acts on A N ( E ) through its translation a c tion on its e lf in the exp o- nent . When A is E -ﬂat, we deﬁne W ﬂ R,E ( A ) to b e the la rgest of the sub- R -a lgebras B ⊆ A N ( E ) having the pr op erties that B is stable under the action of N ( E ) and that the induced action o n B is a Λ R,E -structure. It is elementary to check that a maximal such subalgebra W ﬂ R,E ( A ) exists. THE BASIC GEOMETR Y OF W ITT VECTORS, I 3 This deﬁnition can b e express e d as a universal prop erty . Let Ring ﬂ Λ R,E denote the following categor y: the ob jects are E -ﬂat R -alge br as equipp ed with a Λ R,E - structure, and the morphisms are N ( E ) -equiv arian t R -alg ebra maps. Then W ﬂ R,E , viewed as a functor Ring ﬂ R → R ing ﬂ Λ R,E , is the right adjoint of the evident for getful functor. One then deﬁnes W R,E to b e the left Kan extension of W ﬂ R,E , now viewed a s a functor Ring ﬂ R → Ring R . This amounts to the following. It is no t ha r d to show that the functor W ﬂ R,E is repr esentable, that is , there exists an E -ﬂat R -a lgebra Λ R,E and an isomor phism W ﬂ R,E ( − ) = Hom(Λ R,E , − ), a s set-v alued functor s. Be c ause W ﬂ R,E takes v alues in R -alge bras, Λ R,E carries the structure of a co- R - algebra ob ject in Ring ﬂ R . Becaus e such a structure is describ ed using maps b etw een ce rtain c opro ducts of Λ R,E with itself, and beca use Ring ﬂ R is a full subcateg ory o f Ring R closed under copro ducts, Λ R,E contin ues to be a co- R -algebr a ob ject when viewed a s an ob ject of Ring R . Therefore it repr esents an R -alge bra-v alued functor , and this functor is what W R,E is deﬁned to b e. Since the deﬁnition o f W R,E in terms of W ﬂ R,E is of a purely category- theoretic nature, one sho uld view the E -ﬂat case as the cen tral one. This is in contrast to the common p oint of view that the pur p o se of Witt vector functors is to lift r ings from p ositive characteristic to characteristic zero. As in the E - ﬂat setting, W R,E is the right adjoint of the forgetful functor Ring Λ R,E → Rin g R , but to make sense of this, it is neces sary to know the what a Λ R,E -structure o n a g e ne r al R -algebra is. Unfortunately , it is not easy to s tate the deﬁnition, and so we will leave it to the b o dy o f the pap er . In the E -ﬂat setting, it is equiv alent to a commuting family of F robenius lifts indexed by E , as above; but in g eneral, it is a slight ly strong e r structure tha t is better behaved. When R is Z and E consists o f all maximal ideals of Z , a Λ R,E -structure is equiv alent to a λ -ring structure in the sense of Grothendieck’s Riemann– Ro c h theor y , but this do es not admit a s imple deﬁnition either . In addition to the right adjoint W R,E , the forg etful functor Ring Λ R,E → Ring R has a left adjoint, which we denote b y A 7→ Λ R,E ⊙ A . It ha s a smaller presence in this pap er , but it is v ery impo rtant—ev en in the p -typical case, as the work of Buium [7][8] makes clear. Section 2 de ﬁnes functors W R,E ,n , which ar e trunca tions of W R,E in the s ame wa y that the functors W n ab ov e are trunca tions of W . F or any A ∈ Ring ﬂ R and n ∈ N ( E ) , let W ﬂ R,E ,n ( A ) denote the imag e of the subr ing W ﬂ R,E ( A ) ⊆ A N under the cano nical pro jection A N ( E ) → A [0 ,n ] , where [0 , n ] = { i ∈ N ( E ) | i m 6 n m for all m ∈ E } . Then W ﬂ R,E ,n is a functor Ri ng ﬂ R → Ring ﬂ R . It is r e presentable by an E -ﬂat R -algebr a Λ R,E ,n , and w e extend it to a functor W R,E ,n : Ri ng R → R ing R by ta king its left Kan extension, as ab ove. These trunca ted functors a re related to the or iginal one by the formula W R,E ( A ) = lim n W R,E ,n ( A ) . Even in the p - t ypical case, this approach to deﬁning the Witt vectors has the adv an tage ov er the tr aditional one that univ ersal prop erties a re e mpha sized and the particulars of explicit c o nstructions are played down. But this comes a t a cos t. F or instance, it is not obvious that W R,E ,n preserves surjectivity of maps . T o prov e 4 J. BOR GER this a nd other basic facts, it app ears necess ary to bring back the Witt comp onents ( x 0 , x 1 , . . . ) above, a t least in some for m. T o deﬁne them, the ideals of E mu st be principal; the purp ose o f section 3 is to deﬁne them in the minimal case we will need, which is when E co nsists of a single principa l ideal m . A version o f the pro o f of Witt’s theorem then shows there is a unique functorial bijection A N → W R,E ( A ) such that when A is E - ﬂa t, the compo sition A N → W R,E ( A ) ⊆ A N satisﬁes ( x 0 , x 1 , x 2 , . . . ) 7→ h x 0 , x q 0 + π x 1 , x q 2 0 + π x q 1 + π 2 x 2 , . . . i where q = [ R : m ] a nd π is a ﬁxed gener ator o f m . W e can similar ly iden tify W R,E ,n ( A ) with the quotien t A [0 ,n ] consisting of vectors ( x 0 , . . . , x n ). Let me em- phasize that the compo nents ( x 0 , x 1 , . . . ) depend on the choice of generator π ∈ m in a complex, non-multilinear w ay . But we ca n use them to deﬁne V erschiebung op erator s V j m : m j ⊗ R W R,E ,n ( A ) − → W R,E ,n + j ( A ) π j ⊗ ( x 0 , . . . , x n ) 7→ (0 , . . . , 0 , x 0 , . . . , x n ) , which are indep endent of the choice of the generato r π . Making that so is the purp ose the tensor factor m j . When E consists of a sing le idea l m (possibly nonprincipal), se ction 4 describ es W R,E ,n in ter ms of the case where m is principal, which is covered by section 3. This is done b y w orking Zariski locally on R . Using the same tec hnique, we will show tha t the V erschiebung maps as ab ove can b e deﬁned when m is no t a ssumed to be pr incipal. In fact, there is a unique functoria l family o f such maps ag r eeing with the maps deﬁned ab ov e. The image of V j m is the kernel of the canonical pro jection W R,E ,n + j ( A ) → W R,E ,j ( A ). Similarly , s ection 5 gives a description o f W R,E ,n when E is gener al in ter ms of the case wher e E co nsists of a single ide a l, which is cov ered by section 4: if m 1 , . . . , m r are the ideals in the supp ort of n , then there is a natural isomorphism (0.0.1) W R,E ,n ∼ − → W R, m r ,n m r ◦ · · · ◦ W R, m 1 ,n m 1 . Such a des cription also holds for W R,E , though some car e must b e taken when E is inﬁnite. It is also p ossible to descr ibe the functor Λ R,E ⊙ − , as well as its truncated v a riants Λ R,E ,n ⊙ − , in ter ms o f the case wher e E consists of a single ideal. Section 6 gives several r ing-theore tic facts a b o ut W R,E ,n which w e w ill need later. F or exa mple, this is where we prove that W R,E ,n preserves s urjectivity . Most of the arguments ther e app ear to req uire the use of Witt comp onents and the reduction techn iques of sections 4 a nd 5. Sections 7– 9 prove the ma in r esults, which r elate Witt vector functors and ´ etale maps. Suppos e E consis ts o f a single idea l m . F or any ring A and a ny int eger n > 1, we hav e a diag r am (0.0.2) W R,E ,n ( A ) α n / / W R,E ,n − 1 ( A ) × A s ◦ pr 1 / / t ◦ pr 2 / / A/ m n A. When m is principa l, the maps α n , s , a nd t can b e deﬁned in ter ms of the Witt comp onents re la tive to a ﬁxed gener ator π ∈ m as follows: α n : ( a 0 , . . . , a n ) 7→  ( a 0 , . . . , a n − 1 ) , a q n 0 + π a q n − 1 1 . . . + π n a n  s : ( a 0 , . . . , a n − 1 ) 7→ ( a q n 0 + . . . + π n − 1 a q n − 1 ) mo d m n A t : a 7→ a mo d m n A. If A is m -tor s ion free, (0.0 .2) is an equalizer diagram. Figure 1 sho ws the induced diagram of sch emes in the p -t ypical cas e when n = 1. THE BASIC GEOMETR Y OF W ITT VECTORS, I 5 F rob e nius identity Spec W 1 ( A ) Spec ( A × A ) Spec A/pA Figure 1. As a top ologica l space, Sp ec W 1 ( A ) (traditionally de- noted W 2 ( A )) is tw o copies of Spec A glued alo ng Spec A/pA . This is also true as schemes if we assume that A is p - to rsion free and we glue transversely and with a F robenius twist, as indicated. There is a similar descr iption of Sp ec W n ( A ) as Sp ec W n − 1 ( A ) glued with Spec A along Sp ec A/p n A . See the diagr a m (0 .0 .2). Now le t C denote the following ca tegory: an ob ject is a pa ir ( B , ϕ ), where B is an ´ etale ( W R,E ,n − 1 ( A ) × A )-algebr a and ϕ is an isomorphism of A/ m n A -algebra s A/ m n A ⊗ t ◦ pr 2 B ϕ − → A/ m n A ⊗ s ◦ pr 1 B and where a morphism ( B 1 , ϕ 1 ) → ( B 2 , ϕ 2 ) is a ( W R,E ,n − 1 ( A ) × A )-alg ebra map f : B 1 → B 2 such that ϕ 2 ◦ ( A/ m n A ⊗ t ◦ pr 2 f ) = ( A/ m n A ⊗ s ◦ pr 1 f ) ◦ ϕ 1 . In other words, C is the ca tegory of a lgebras equipp ed with g luing data relative to the dia gram (0 .0 .2), o r eq uiv a lently C is the (weak) ﬁb er product of the catego ry of ´ etale W n − 1 ( A )-algebras and the catego r y ´ etale A -alg ebras ov er the catego ry of ´ etale A/ m n A -algebra s via the evident functor s. Theorem A. The b ase-cha nge functor fr om the c a te gory of ´ etal e W R,E ,n ( A ) -algebr as to C is an e quivalenc e. If A is m -torsion fr e e , then a quasi-inverse is given by send- ing ( B , ϕ ) to the e qualizer of the two maps B 1 ⊗ id B / / ϕ ◦ (1 ⊗ id B ) / / A/ m n A ⊗ s ◦ pr 1 B . The ﬁrst statement can b e expressed succinctly in geo metric terms; it says that the ma p α n satisﬁes eﬀectiv e descent for ´ etale algebras and that des c e nt da ta is equiv alent to gluing data with r esp ect to the dia gram (0.0.2). Using theorem A a nd induction on n , it is in principle poss ible to reduce questions about ´ etale W n ( A )- algebras to questions abo ut ´ etale A -a lgebras. This is still true when E cons ists of more than o ne idea l, but now by (0.0.1) and induction on r . Section 9 g eneralizes v an der Kalle n’s theor em [2 4], (2.4), to arbitr a ry R and E : Theorem B. L et f : A → B b e an ´ etale m orphism of R -algebr as. Then the map W R,E ,n ( f ) : W R,E ,n ( A ) → W R,E ,n ( B ) of R -algebr as is ´ etale. This theorem is fundamental in extending W itt constructions b eyond aﬃne schemes a nd will b e use d often in [4]. V an der K allen’s arg ument , which has an 6 J. BOR GE R inﬁnitesimal ﬂav or , co uld b e extended to our setting with only minor mo diﬁca tions. Instead we deduce theorem B from theor em A, a nd so o ur argument has a glob- ally geometr ic ﬂa vor. (Note that, until recently , v an der K allen’s pap er [2 4] had escap ed the notice of ma ny workers in de Rham–Witt theory , to whom theore m B was unknown even for the p -typical Witt vectors.) Last, I would like to thank Amnon Neeman for helpful discus sions o n some techn ical p o int s and La nce Gurney for some comment s on earlie r versions o f this pap er. Contents Int ro duction 1 1. Generalized Witt vectors and Λ-rings 6 2. Grading a nd truncations 16 3. Principal s ingle-prime case 19 4. General sing le-prime cas e 22 5. Multiple-prime ca se 25 6. Basic aﬃne prope r ties 26 7. Some g eneral descent 29 8. Ghost des c ent in the single-prime case 35 9. W a nd ´ etale morphisms 38 References 41 1. Generalized Witt vectors and Λ -rings The pur po se of this section is to deﬁne o ur genera liz ed Witt vectors and Λ- r ings. It is larg ely an expansion in more concrete terms of Borger– Wieland [5], o r r ather of the parts ab out Witt vectors and Λ-r ing s. The appr oach here will allow us to av oid muc h of the abs tr act langua g e of op era tions on r ing s, as ﬁr st intro duce d in T all–W raith [23]. F o r the traditional w ay of de ﬁning Λ-r ings and Witt vectors, s e e § 1 of chapter IX o f Bour baki [6] and e s pe cially the exerc is es for that section. O ne ca n also see Witt’s original pap er [26] on the p -typical Witt vectors (r eprinted in [27]) and his notes on the big Witt v e c to rs [27], pp. 157–1 63. 1.1. Supr amaximal ide als. Let us s ay that an idea l m of a r ing R is supra ma ximal if either (a) R / m is a ﬁnite ﬁeld, R m is a discr e te v aluation ring, and m is ﬁnitely pre- sented as a n R -mo dule, or (b) m is the unit ideal. By far the most imp o rtant example is a max imal ideal with ﬁnite residue ﬁeld in a Dedekind domain. (In fact, all phenomena in this pa p er o ccur a lready ov er R = Z , and this case covers the classic a l Witt v ector s and λ -rings.) The rea son we allow the unit ideal is only so that a supra maximal ideal r emains s upramaximal after a ny lo calization. Note that a suprama x imal ideal m is inv er tible as a n R -mo dule. Indee d, lo cally at m it is the max imal ide a l of a discre te v alua tion ring, and awa y from m it is the unit ideal. THE BASIC GEOMETR Y OF W ITT VECTORS, I 7 1.2. Gener al n otation. F ix a ring R and a fa mily ( m α ) α ∈ E of pairwise coprime supramaximal ideals of R indexed b y a set E . Note tha t b eca use the unit ide a l is coprime to itself, it can be rep eated an y num b er of times; otherwise the ideals m α are distinct. F or each α ∈ E , let q α denote the cardinality of R/ m α . W e will often abusively s pe ak of m α rather than α a s b eing a n element of E , e sp ecially when m α is maximal, in which ca se it comes from a unique α ∈ E . Let R [1 / E ] denote the R -algebr a whose sp ectrum is the c omplement o f E in Spec R . It is the univ e r sal R -a lgebra in which every m α bec omes the unit ideal. It also has the more concr ete description R [1 /E ] = O α ∈ E R [1 / m α ] , where the tenso r pro duct is over R a nd R [1 / m α ] is deﬁned to b e the co equa lizer of the maps Sym( R ) / / / / Sym( m − 1 α ) of symmetric algebr as, where m − 1 α is the dual of m α , one of the ma ps is Sym a pplied to the canonica l map R → m − 1 α , and the other is the map induced by the R - mo dule map R → Sym( m − 1 α ) that sends 1 ∈ R to the element 1 ∈ Sym( m − 1 α ) in degree z ero. Finally , we write N for the monoid { 0 , 1 , 2 , . . . } under addition and write Ri ng R for the category of R -algebr as. 1.3. E -ﬂat R -mo dules. Let us say that an R -mo dule M is E -ﬂat if for all maximal ideals m in E , the follo wing equiv a lent conditions a re sa tisﬁed: (a) R m ⊗ R M is a ﬂat R m -mo dule, (b) the map m ⊗ R M → M is injective. The equiv alence of these tw o can be seen as follows. Condition (b) is equiv a le nt to the statement T o r R 1 ( R/ m , M ) = 0, which is eq uiv a lent to T or R m 1 ( R/ m , R m ⊗ R M ) = 0 . Since R m is a discr ete v a luation ring, this is e quiv a lent to the R m -mo dule R m ⊗ R M being torsio n free and hence ﬂa t. W e say an R -alg e bra is E -ﬂat if its underlying R -mo dule is. Let Ring ﬂ R denote the full subcateg ory of Ring R consisting o f the E -ﬂat R -algebr as. 1.4. Prop ositio n. Any pr o duct of E -ﬂat R -mo dules is E -ﬂat, and any su b- R - mo dule of an E -ﬂat R -mo dule is E -ﬂat. Pr o of. W e will use condition (b) ab ov e. Le t ( M i ) i ∈ I be a family of E -ﬂat R -mo dules. W e wan t to s how that for each ma ximal ideal m in E , the comp o sition m ⊗ Q i M i / / Q i m ⊗ M i / / Q i M i is injective. Beca use each M i is E -ﬂat, the rig ht-hand map is injective, and so it is enough to s how the left-hand map is injective. Since m is assumed to b e ﬁnitely presented as an R -mo dule, we can express it as a cokernel of a map N ′ → N of ﬁnite free R -mo dules. Then we hav e the following diagram with exact rows N ′ ⊗ R Q i M i / / ∼   N ⊗ R Q i M i / / ∼   m ⊗ R Q i M i / /   0 Q i N ′ ⊗ R M i / / Q i N ⊗ R M i / / Q i m ⊗ R M i / / 0 . The left t wo vertical ma ps are isomor phis ms b ecause N ′ and N are ﬁnite free. Therefore the right most vertical ma p is an injection (and even an isomorphism). 8 J. BOR GE R Now supp ose M ′ is a sub- R -mo dule of an E -ﬂat R -mo dule M . Since m is an inv ertible R -mo dule, m ⊗ R M ′ maps injectively to m ⊗ R M . Since M is E -ﬂat, m ⊗ R M ′ further ma ps injectively to M , and hence to M ′ .  1.5. Ψ -rings. Let A b e an R -alg ebra. L e t us deﬁne a Ψ R,E -action, or a Ψ R,E -ring structure, on A to b e a commuting family of R - algebra endomorphisms ψ α indexed by α ∈ E . This is the same as an action of the monoid N ( E ) = L E N on A . F or any e lement n ∈ N ( E ) , w e will a lso write ψ n for the endomorphism of A induced by n . A morphism of Ψ R,E -rings is deﬁned to b e a n N ( E ) -equiv ariant mor phism of rings. The free Ψ R,E -ring on one gener a tor e is Ψ R,E = R [ e ] ⊗ R N ( E ) , where N ( E ) acts on Ψ R,E through its action o n itself in the exp onent. In pa rticular, Ψ R,E is freely generated a s a n R -algebra by the elements ψ n ( e ), wher e n ∈ N ( E ) . Then it is natural to write ψ n = ψ n ( e ) ∈ Ψ R,E and ψ α = ψ b α ∈ Ψ R,E , where b α ∈ N ( E ) denotes the α -th s tandard basis v ector, and e = ψ 0 ∈ Ψ R,E for the ident ity op erator . F o r any Ψ R,E -ring A , there is a unique set map (1.5.1) Ψ R,E × A ◦ − → A with the prop e rty that for all α ∈ E , r ∈ R , f 1 , f 2 ∈ Ψ R,E , a ∈ A we hav e (1.5.2) ψ α ◦ a = ψ α ( a ) and (1.5.3) r ◦ a = r, ( f 1 + f 2 ) ◦ a = ( f 1 ◦ a ) + ( f 2 ◦ a ) , ( f 1 f 2 ) ◦ a = ( f 1 ◦ a )( f 2 ◦ a ) . T aking A = Ψ R,E , we get a bina ry op era tio n ◦ o n Ψ R,E called c omp osition or plethysm . One can check that this makes Ψ R,E a mo noid (nonco mmut ative unless R = 0 ) with iden tity e a nd that (1 .5.1) is a monoid a c tion. In the language of pleth y s tic algebra [5], we can in terpr et Ψ R,E as the free R - plethory R h ψ α | α ∈ E i on the R - algebra endomorphisms ψ α . Then a Ψ R,E -action in the sense ab ove is the same as a Ψ R,E -action in the sense of abs tract plethystic algebra. In pa rticular, Ψ R,E can b e viewed as the ring of natural unary o p e ra- tions o n Ψ R,E -rings, a nd the comp osition op eration ◦ ab ov e agre e s with the usual comp osition of unary op eratio ns . (Compare with 1.18 b elow) 1.6. E -ﬂat Λ -rings. Let A b e an R -algebr a which is E -ﬂat. Deﬁne a Λ R,E -action, or a Λ R,E -ring structure, on A to b e a Ψ R,E -action with the following F r ob enius lift prop erty: for all α ∈ E , the endomor phism id ⊗ ψ α of R/ m α ⊗ R A ag r ees with the F r ob enius map x 7→ x q α . A morphism of E -ﬂat Λ R,E -rings is simply deﬁned to b e a morphism of the underlying Ψ R,E -rings. Let us denote this categor y by Ring ﬂ Λ R,E . 1.7. The ghost ring. Since an action of Ψ R,E on an R -algebr a A is the same a s an action (in the c a tegory of R -a lg ebras) of the monoid N ( E ) , the forg e tful functor from the category of Ψ R,E -rings to that of R -algebr as has a right a djoint given by A 7→ Y N ( E ) A = A N ( E ) , where N ( E ) acts on A N ( E ) through its action o n itself in the exponent. (This is a general fact ab out monoid actions in any categ ory with pro ducts.) F or a ∈ A N ( E ) and n, n ′ ∈ N ( E ) , the n -th comp onent of ψ n ′ ( a ) is the ( n + n ′ )-th comp onent of a . One might call A N ( E ) the co free Ψ R,E -ring on the R -alg ebra A . It has tradition- ally bee n called the ring of ghost c omp onents or ghost ve ctors . By 1.4, it is E -ﬂat if A is. When | E | = 1, there is the p o s sibility of confusing the ghost ring A N , which has the pro duct ring structure, with the usual ring A N of Witt comp onents (see THE BASIC GEOMETR Y OF W ITT VECTORS, I 9 3.5), which has an exotic ring s tructure. T o pr even t this, we will use angle br ack ets h a 0 , a 1 , . . . i for elemen ts of the g host ring. 1.8. Witt ve ctors of E -ﬂat rings. Let us now construct the functor W ﬂ R,E . W e will show in 1.9 that it is the rig ht adjoint of the for getful functor from the category of E -ﬂat Λ R,E -rings to that of E -ﬂat R -a lg ebras. (F urther, the ﬂatness will b e remov ed in 1.12.) Let A b e a E -ﬂat R -a lgebra. Let U 0 ( A ) deno te the cofre e Ψ R,E -ring A N ( E ) . F or any i > 0, let U i +1 ( A ) = { b ∈ U i ( A ) | ψ α ( b ) − b q α ∈ m α U i ( A ) for a ll α ∈ E } . This is a sub- R -alg ebra of A N ( E ) . Indeed, it is the in tersection over α ∈ E of the equalizers of pairs of R -alge bra maps U i ( A ) / / / / R/ m α ⊗ R U i ( A ) given by x 7→ 1 ⊗ ψ α ( x ) and b y x 7→ (1 ⊗ x ) q α . Now deﬁne (1.8.1) W ﬂ R,E ( A ) = \ i > 0 U i ( A ) . This is the ring o f E -typic al Witt ve ctors with entries in A . It is a s ub- R -alg ebra of A N ( E ) . Obse r ve that W ﬂ R,E ( A ) = A N ( E ) if A is an R [1 / E ]-algebra. 1.9. Prop ositio n. (a) W ﬂ R,E ( A ) is a sub- Ψ R,E -ring of A N ( E ) . (b) This Ψ R,E -ring structu r e on W ﬂ R,E ( A ) is a Λ R,E -ring structu r e. (c) The induc e d functor A 7→ W ﬂ R,E ( A ) fr om E -ﬂat R -algebr as to E - ﬂ at Λ R,E - rings is the right adjoi nt of the for getful functor. Pr o of. (a): Let us ﬁrst show by induction that each U i ( A ) is a sub-Ψ R,E -ring of A N ( E ) . F o r i = 0, w e hav e U 0 ( A ) = A N ( E ) , and so it is clear . F or i > 1 , we use the description o f U i +1 ( A ) as the intersection of the e q ualizers of the pairs of ring maps U i ( A ) / / / / R/ m α ⊗ R U i ( A ) given in 1.8. Observe that bo th these ring maps be c o me Ψ R,E -ring maps if we give R/ m α ⊗ R U i ( A ) a Ψ R,E -action by deﬁning ψ β : a ⊗ x 7→ a ⊗ ψ β ( x ), for all β ∈ E . Since limits of Ψ R,E -rings exist and their underlying rings agre e with the limits taken in the ca tegory of ring s, U i +1 ( A ) is a sub- Ψ R,E -ring of A N ( E ) . Ther efore W ﬂ R,E ( A ), the in tersection o f the U i ( A ), is also a sub-Ψ R,E -ring of A N ( E ) . (b): It is enough to v e r ify ψ α ( x ) − x q α ∈ m α W ﬂ R,E ( A ) = m α \ i > 0 U i ( A ) , for all α ∈ E and x ∈ W ﬂ R,E ( A ). F or an y i > 0, we k now ψ α ( x ) − x q α ∈ m α U i ( A ) , bec ause x ∈ W ﬂ R,E ( A ) ⊆ U i +1 ( A ). Ther efore we know ψ α ( x ) − x q α ∈ \ i > 0 m α U i ( A ) . So, it is enough to show (1.9.1) m α \ i > 0 U i ( A ) = \ i > 0 m α U i ( A ) . 10 J. BOR GE R Since m α is ﬁnitely g enerated, it is a q uotient o f a ﬁnite free R -mo dule N . Consider the induced diagram m α ⊗ R lim i U i ( A ) h / / lim i m α ⊗ R U i ( A ) N ⊗ R lim i U i ( A ) f / / O O lim i N ⊗ R U i ( A ) g O O Since N is ﬁnite free, f is an isomor phism; since m α is pro jective, the map N → m α has a section and hence so do es g . Therefore g ◦ f is surjective and hence so is h , which implies (1.9.1). (c): Let A b e an E -ﬂat R -alg ebra, let B b e an E -ﬂat Λ R,E -ring, a nd let ¯ γ : B → A be a n R -algebr a map. By the co free pr o p erty of A N ( E ) , there is a unique Ψ R,E -ring map γ : B → A N ( E ) lifting ¯ γ . W e now only need to show that the image of γ is contained in W ﬂ R,E ( A ). By induction, it is e nough to show that if im( γ ) ⊆ U i ( A ), then im( γ ) ⊆ U i +1 ( A ). Let b be an element o f B . Then for each α ∈ E , w e ha ve ψ α  γ ( b )  − γ ( b ) q α = γ  ψ α ( b ) − b q α  ∈ γ ( m α B ) ⊆ m α im( γ ) ⊆ m α U i ( A ) . Therefore, by deﬁnition of U i +1 ( A ), the e le ment γ ( b ) lies in U i +1 ( A ).  1.10. Exer cises. Let R = Z . If E co nsists of the single ideal p Z , then W ﬂ ( Z ) agrees with the subr ing o f the gho st ring Z N consisting of v ecto rs a = h a 0 , a 1 , . . . i that sa tisfy a n ≡ a n +1 mo d p n +1 for all n > 0. In particular, the elemen ts a re p -adic Cauch y seq uences and the rule a 7→ lim n →∞ a n deﬁnes a surjective ring map W ﬂ ( Z ) → Z p . W e can go a step further with W ﬂ ( Z p ). Let I denote the ideal p Z p × p 2 Z p × · · · in Z N p . Then W ﬂ ( Z p ) is isomorphic to the ring Z p ⊕ I with multiplication deﬁned by the form ula ( x, y )( x ′ , y ′ ) = ( xx ′ , xy ′ + y x ′ + y y ′ ). Now supp os e that E co nsists of all the maximal idea ls of Z , and identify N ( E ) with the set of po sitive integers, by unique factorization. Then W ﬂ ( Z ) consists of the gho s t vectors h a 1 , a 2 , . . . i that satisfy a j ≡ a pj mo d p 1+ord p ( j ) for all j > 1 and all pr imes p . 1.11. R epr esen ting W ﬂ . Let us co nstruct a ﬂat R -algebr a Λ R,E representing the functor W ﬂ R,E . Fir st we will co nstruct ob jects Λ i R,E representing the functor s U i . F o r i = 0, it is clear : U 0 is re presented by Λ 0 R,E = Ψ R,E . Now ass ume Λ i R,E has bee n cons tructed and that it is a sub- R -a lgebra of R [1 /E ] ⊗ R Ψ R,E satisfying R [1 /E ] ⊗ R Λ i R,E = R [1 /E ] ⊗ R Ψ R,E . Then let Λ i +1 R,E denote the sub-Λ i R,E -algebra of R [1 /E ] ⊗ R Ψ R,E generated by all elements π ∗ ⊗ ( ψ α ( f ) − f q α ), where π ∗ ∈ m − 1 α ⊆ R [1 /E ], f ∈ Λ i R,E , a nd α ∈ E . Then Λ i R,E is ﬂat ov er R . Indeed, it is E -ﬂat b eca use it is a s ub- R -a lg ebra o f R [1 /E ] ⊗ R Ψ R,E , and it is ﬂat awa y from E b ecause R [1 /E ] ⊗ R Λ R,E agrees with the free R [1 /E ]-algebr a R [1 /E ] ⊗ R Ψ R,E . It also clear ly r epresents U i . Finally , we set (1.11.1) Λ R,E = [ i > 0 Λ i R,E ⊆ R [1 /E ] ⊗ R Ψ R,E . THE BASIC GEOMETR Y OF W ITT VECTORS, I 11 It is ﬂat over R b ecause it is a colimit of ﬂat R -algebra s, and it represents W ﬂ R,E bec ause e ach Λ i R,E represents U i . As a n exa mple, if E = E ′ ∐ E ′′ , where E ′′ con- sists of o nly copies of the unit ideal, then Λ R,E agrees with the monoid algebra Λ R,E ′ [ N ( E ′′ ) ]. W e will often use the shortened forms Λ E or, when E = { m } , Λ m . Since Λ R,E represents W ﬂ , which takes v alues in R -algebra s, Λ R,E carries the structure o f a co - R -a lgebra ob ject in Ring ﬂ R . Because Rin g ﬂ R is c lo sed under copro d- ucts (the tensor pr o duct o f ﬂat modules b eing ﬂa t), a co-ring structur e consists in morphisms (1.11.2) ∆ + , ∆ × : Λ R,E − → Λ R,E ⊗ R Λ R,E , ε + , ε × : Λ R,E − → R corres p o nding to addition, multiplication, the additive identit y , and the multiplica- tive identit y o n the functor W ﬂ R,E . The R - line a r structure on W ﬂ R,E corres p o nds to a morphism (1.11.3) β : Λ R,E → R R = Y R R. All these s tructure maps sa tis fy the opp osite of the R -a lgebra a xioms. (In the language of schemes, one would say this makes Spec Λ R,E an R -algebr a scheme ov er R ; or in the lang uage of [5], it makes Λ R,E an R - R -biring .) 1.12. Deﬁn ition of W in gener al. W e can view Λ R,E as an o b ject o f Ring R , instea d of Ring ﬂ R . Then deﬁne W R,E as a s et-v alued functor on Ring R by (1.12.1) W R,E ( A ) = Hom Ring R (Λ R,E , A ) . The structure maps (1 .11.2)–(1.11.3) give W R,E the structure of a functor with v a lues in R -algebra s: (1.12.2) W R,E : Ri ng R − → Ring R . (Note that here we really use the fact the the copro duct in Ring ﬂ R agrees with that in Rin g R . In 1.11, it was used only to justify the symbol ⊗ for the copro duct.) F o r any A ∈ Ring R , let us ca ll the W R,E ( A ) the R -algebr a of E -typic al Witt ve ctors with entries in A . Its r estriction to Ring ﬂ R agrees with W ﬂ R,E bec ause Ri ng ﬂ R is a full sub categor y of Ring R . W e will often write W E or W for W R,E when ther e is no risk of c o nfusion. When E consists of a single ideal m , we will also write W R, m or W m . 1.13. R emark: Kan extensions. In ca tegorica l terms, W R,E is the left Kan exten- sion of i ◦ W ﬂ R,E along the inc lus ion functor i : (1.13.1) Ring ﬂ R i / / Ring R Ring ﬂ R W fl R,E O O i / / Ring R . W R,E O O ✤ ✤ ✤ (See Bor ceux [2], 3.7, for example, for the genera l theory of Kan ex tensions.) I men tion this only to emphasize tha t the passage fr o m the E -ﬂat cas e to the ge ner al case is by a purely catego ry-theore tic pro cess, and hence the heart o f the theory lies in the E -ﬂat case. This is in contrast to the c ommon p oint o f view that the purp o s e of Witt vector functors is to lift rings fro m p ositive characteristic to characteris tic zero. 12 J. BOR GE R 1.14. Ghost map w . The ghost map w : W R,E ( A ) − → Y N ( E ) A is the natura l map induce d b y the universal pr op erty of Ka n extens ions applied to the inclusion maps W ﬂ R,E ( A ) → Q N ( E ) A , which ar e functorial in A . Equiv a lent ly , it is the morphism of functor s induced b y the map Ψ R,E = Λ 0 R,E − → Λ R,E of representing ob jects. When A is E -ﬂat, it is har mless to identify w with the inclusion map. 1.15. Example: p -t ypic al and big Witt ve ctors. Supp ose R is Z . If E consists of the single ideal p Z , then W agree s with the classical p -typical Witt vector functor [2 6]. Indeed, for p -torsion free rings A , this fo llows fro m Ca rtier’s lemma, whic h says that the traditionally deﬁned p -typical Witt vector functor res tricted to the catego ry o f p -torsio n-free r ings ha s the same universal pr op erty as W ﬂ . (See B o urbaki [6], IX.44, exercise 14 or Laz ard [20], VII § 4.) T he r efore, they are isomorphic functors. F o r A g e neral, one just observes that that the traditional functor is repres ented by the ring Z [ x 0 , x 1 , . . . ], which is p -tors ion free, and so it is the left Kan ex tension of its restric tion to the category of p - torsion- fr ee rings . Therefore it ag rees with W as deﬁned here . Another pr o of o f this is given in 3.5. It makes a direct co nnection with the traditional Witt compo nent s, rather than going thro ugh the univ er sal prop erty . Suppo se instead that E is the family of all maximal ideals o f Z . Then W ag r ees with the class ical big Witt vector functor. As ab ov e, this can b e shown by reducing to the tor s ion-free ca se and then citing the analogue of Car tier’s lemma. (Whic h version of Ca r tier’s lemma dep ends on ho w we de ﬁne the classica l big Witt vector functor. If we use ge ne r alized Witt p o lynomials, then w e need Bo urbaki [6 ], IX.55, exercise 4 1b. If it is deﬁned as the c ofree λ -ring functor, as in Grothendieck [11], then we ne e d Wilkerson’s theorem [2 5], prop osition 1 .2.) Finally , we will see be low (3.5) that when R is a complete discr ete v alua tion ring and E consists of the maxima l ideal of R , then W agrees with Haz e winkel’s ramiﬁed Witt vector functor [1 6], (18.6 .1 3). 1.16. Comonad struct u r e on W . The functor W ﬂ : Ring ﬂ R → Rin g ﬂ R is naturally a como nad, b eing the compo sition o f a functor (the forg etful one ) with its right adjoint, and this comonad structure prolongs naturally to W R,E . The r eason for this can b e expres sed in tw o ways—in ter ms of Kan extensio ns or in terms of representing ob jects. The ﬁrs t way is to invok e the g eneral fact that W R,E , as the Kan extension of the co monad W ﬂ R,E , has a natura l comonad structur e . This us e s the co mm utativity of (1.1 3.1) and the fullness and faithfulness of i . The other wa y is to trans la te the structure on W ﬂ of being a comonad into a structure on its re presenting ob ject Λ R,E . One then obs erves that this is exactly the s tr ucture for the underlying R - algebra i (Λ R,E ) to repre sent a comonad on Rin g R . (This is called an R -plethory structure in [5].) 1.17. Λ -r ings. The category R ing Λ R,E of Λ R,E -rings is by deﬁnition the categor y of coalgebras for the comonad W R,E , that is, the categor y of R -alg ebras equipp ed with a coactio n of the comona d W R,E . Since W R,E extends W ﬂ R,E , a Λ R,E -ring structure o n a n E -ﬂat R -a lgebra A is the same as a co mm uting family o f F ro b enius lifts ψ α . THE BASIC GEOMETR Y OF W ITT VECTORS, I 13 When R = Z and E is the family of a ll maxima l ideals of Z , then a Λ-r ing is the same as a λ -ring in the sense of Grothendieck’s Riemann–Ro ch theory [11] (and origina lly calle d a “sp ecial λ -ring”). In the E -ﬂat case, this is Wilkerson’s theorem ([25], prop os ition 1.2). The proo f is an exer cise in symmetric functions, but the dee p er meaning eludes me. The general case follows from the E - ﬂat cas e by catego ry theory , as in 1.15. 1.18. F r e e Λ - rings and Λ ⊙ − . Since W R,E is a representable co monad o n Ring R , the for getful functor fro m the catego ry of Λ R,E -rings to the categ ory of R -alge bras has a left adjoin t denoted Λ R,E ⊙ − . This follows either from the adjoint functor theorem in category theory (3.3.3 of Bor ceux [2]), or b y simply w r iting down the adjoint in terms of generator s and r e lations, as in 1.3 o f Borger –Wieland [5]. The second approach involv es the R -plethory structure on Λ R,E , and is similar to the de- scription of tensor pro ducts, fr ee diﬀerential rings, and so o n in terms o f ge ne r ators and relatio ns. The functor Λ R,E ⊙ − , viewed a s a n endofunctor o n the catego r y of R -a lgebras , is naturally a monad, simply b eca use it is the left adjoint of the comonad W R,E . F ur ther, the catego ry of algebr as for this monad is natur ally equiv alent to Ring Λ R,E . This can b e proved using Beck’s theorem (4.4.4 of Borce ux [3]), and is the same as the fact that the catego ry of K - mo dules, for any r ing K , can b e deﬁned as the category of a lgebras for the monad K ⊗ − or coalg ebras for the comonad Hom( K, − ). W e can interpret ele ment s of Λ R,E as natural op era tio ns o n Λ R,E -rings. Indeed, a Λ R,E -ring structure on a ring A is by deﬁnition a (t yp e o f ) map A → W R,E ( A ). It therefor e induces a set map Λ R,E × A − → Λ R,E × W R,E ( A ) = Λ R,E × Hom R (Λ R,E , A ) − → A, which is functorial in A . In pa rticular, if w e take A = Λ R,E , we get a set map (1.18.1) Λ R,E × Λ R,E ◦ − → Λ R,E . It agrees with the restriction of the compo sition map ◦ o n Ψ R [1 /E ] ,E = R [1 / E ] ⊗ R Ψ R,E given in 1.5. In particular, it is asso ciative with ide ntit y e . In fact, all natural op era tio ns on Λ R,E -rings come from Λ R,E in this wa y . See [5] for a n a bs tract account fro m this p o int of vie w . 1.19. R emark: identity-b ase d appr o aches. It is p os sible to se t up the theory of Λ R,E - rings mor e concretely using universal iden tities rather than category theory . (See Buium, Buium– Sima nca, and Joyal [7][9][18][19], for example.) In this subse c tion, I will say something a bo ut that p oint o f v iew and its r elation to the ca tegory-theo retic one, but it will not be used elsewhere in this paper. First supp ose that for each α ∈ E , the idea l m α is gener ated by a s ingle element π α . F or any Λ R,E -ring A and any element a ∈ A , ther e ex ists an element δ α ( a ) ∈ A such that ψ α ( a ) = a q α + π α δ α ( a ) . If w e now assume that A is E -ﬂat, then the element δ α ( a ) is uniquely determined by this equation, and therefore δ α deﬁnes a n op erator o n A : δ α ( a ) = ψ α ( a ) − a q α π α . Observe that if the integer q α maps to 0 in R , for exa mple when R is a r ing of int egers in a function ﬁe ld, then δ α is additive; but other wis e it essentially never is. (Also no te that δ α is the s ame as the op erator θ π α , 1 deﬁned in 3.1 below.) Conv ers e ly , if A is an E -ﬂat R -algebr a, equippe d with with o pe rators δ α , then there is at most o ne Λ R,E -ring s tructure o n A whose δ α -op erator s a re the given ones. T o sa y when suc h a Λ R,E -ring structure exists, we only nee d to express in terms 14 J. BOR GE R of the op erator s δ α the c ondition that the oper ators ψ α be commuting R -a lgebra homomorphisms. After dividing b y any acc umulated factor s o f π α , this giv e s the ident ities o f Buium–Simanca [9], deﬁnition 2.1: δ α ( r ) = r − r q α π α , for r ∈ R, (1.19.1) δ α ( a + b ) = δ α ( a ) + δ α ( b ) + C α ( a, b ) , (1.19.2) δ α ( ab ) = δ α ( a ) b q α + a q α δ α ( b ) + π α δ α ( a ) δ α ( b ) , (1.19.3) δ α ◦ δ α ′ ( a ) = δ α ′ ◦ δ α ( a ) + C α,α ′  a, δ α ( a ) , δ α ′ ( a )  , (1.19.4) where (1.19.5) C α ( x, y ) = x q α + y q α − ( x + y ) q α π α = − q α − 1 X i =1 1 π α  q α i  x q α − i y i , (1.19.6) C α,α ′ ( x, y , z ) = C α ′ ( x q α , π α y ) π α − C α ( x q α ′ , π α ′ z ) π α ′ − δ α ( π α ′ ) π α ′ z q α + δ α ′ ( π α ) π α y q α ′ . One can ea sily chec k that the coeﬃcients of these polyno mials ar e ele men ts o f R . F o r a ny R -alge bra A , let us deﬁne a δ R,E -structure on A to be a family o f op- erators δ α satisfying the axio ms ab ov e . Thus, if A is an E - ﬂat R -algebr a , then a Λ R,E -structure—by deﬁnition a comm uting fa mily o f F r o b enius lifts indexe d by E — is equiv alent to a δ R,E -structure. The p o int o f all this, then, is that if we no longer require A to b e E -ﬂat, a δ R,E -structure is genera lly str onger than having a com- m uting family of F r ob enius lifts, but it is still equiv alent to ha ving a Λ R,E -structure. This oﬀers ano ther p oint of v iew on the diﬀerence be t ween a Λ R,E -structure and a commuting family of F rob enius lifts: A δ R,E -structure is w ell b ehaved fro m the po int of view of univ ersal algebra (and hence so is a Λ R,E -structure) b ecause it is given b y o pe rators δ α whose eﬀect on the ring str ucture is describe d b y universal ident ities, a s ab ove; but the structure of a commuting family of F rob enius lifts do es not have this pro p erty b ecause o f the existential quantiﬁer hidden in the w o rd lift . The equiv alence b etw een δ R,E -structures and Λ R,E -structures can b e s een as follows. F or E -ﬂat R -alg ebras A , it was explained ab ove. F or general A , the e quiv- alence can b e shown by c hecking that the cofree δ R,E -ring functor is repre s ented by an E -ﬂat R - algebra (in fact, a free one). It therefore a g rees with the left Ka n extension o f its restriction to the categ ory of E -ﬂat a lg ebras, and hence ag rees with W R,E . W e co uld extend the iden tity-based approa ch to the ca se where the idea ls m α are not principal, but then we would need op er ators (1.19.7) δ α,π ∗ α ( x ) = π ∗ α ( ψ α ( x ) − x q α ) for every element π ∗ α ∈ m − 1 α , o r at least for those in a chosen g e nerating set o f m − 1 α , and we would need additio nal axioms relating them. A particular ly co nv enient generating set of m − 1 α is one of the fo rm { 1 , π ∗ α } , which a lwa ys exists. F urther, for e a ch α ∈ E , it is enough to use the op erators ψ α and δ α,π ∗ α instead o f δ α, 1 and δ α,π ∗ α , b e cause δ α, 1 can be e xpressed in terms of ψ α , by (1.19.7). Therefore if we ﬁx elements π ∗ α ∈ m − 1 α which ar e R -mo dule generato rs mo dulo 1 , the relations needed for the gener ating set S α ∈ E { ψ α , δ α,π ∗ α } of op erators ar e those in (1.19.1)– (1.19.6) but one needs to make the following changes: for e a ch α ∈ E , repla c e ea ch o ccurrence of π − 1 α with π ∗ α , and a dd a xioms that ψ α is a n R -alg ebra homomorphism, that ψ α commutes with all ψ α ′ and all δ α ′ ,π ∗ α ′ , and that (1.1 9 .7) holds. THE BASIC GEOMETR Y OF W ITT VECTORS, I 15 When R is an F p -algebra for s ome prime num b er p , the p olynomials C α ( x, y ) are zer o and the axioms a b ove s implify consider ably . In particular, the op era tors δ α are additive, and so it is p o ssible to describ e a Λ R,E -structure using a co c ommu- tative twisted bia lgebra, the additiv e bialgebra o f the plethor y Λ R,E . (See Bo rger– Wieland [5], sections 2 a nd 10.) 1.20. L o c alization of the ring R of sc alars. Let R ′ be an E -ﬂat R -algebr a such that the structur e ma p R → R ′ is an epimorphis m of rings. (F or exa mple, the map Spec R ′ → Sp ec R could b e an op en immer sion.) Then the family ( m α ) α ∈ E induces a family ( m ′ α ) α ∈ E of ideals of R ′ , where m ′ α = m α R ′ . By the assumptions on R ′ , each m ′ α is supramaxima l. Let us write E ′ = E and use the notation E ′ for the index set of the m ′ α . Let us c onstruct an isomorphism: (1.20.1) R ′ ⊗ R Λ R,E ∼ − → Λ R ′ ,E ′ . The category Ri ng ﬂ Λ R ′ ,E ′ (see 1.6) is a subc ategory of the categ ory o f Ring ﬂ Λ R,E . Indeed, an y ob ject A ′ ∈ Ring ﬂ Λ R ′ ,E ′ is an R -alg ebra with endomor phisms ψ m α , for each α ∈ E . These endomorphisms are again commuting F rob enius lifts, simply bec ause A ′ / m ′ α A ′ = A ′ / m α A ′ . Since A ′ is E ′ -ﬂat (a nd b y the assumptions o n R ′ ), A ′ is E -ﬂat. Ther efore, it can b e v ie wed as a Λ R,E -ring. F ur ther, Ri ng ﬂ Λ R ′ ,E ′ agrees with the sub catego ry of Ring ﬂ Λ R,E consisting of ob jects A whose structure map R → A factors through R ′ , necess arily uniquely . Now co n- sider the underlying- set functor on this categor y . F rom the deﬁnition of Ring ﬂ Λ R ′ ,E ′ , this functor is r e presented by the rig h t-hand s ide of (1 .20.1), and from the second description, it is represented by the le ft-ha nd side. Let (1.20.1) be the induced isomorphism on representing ob jects. It sends an elemen t r ′ ⊗ f to r ′ f . The is o morphism of represented functors which is induced b y (1.20 .1) gives na t- ural maps (1.20.2) W R ′ ,E ′ ( A ′ ) ∼ − → W R,E ( A ′ ) , for R ′ -algebra s A ′ . Finally , let us show that for any R ′ -algebra B ′ , the following cano nical map is an isomor phism: (1.20.3) Λ R,E ⊙ B ′ ∼ − → Λ R ′ ,E ′ ⊙ B ′ . It is eno ugh to show tha t for an y R ′ -algebra A ′ , the induced map Hom R ′ (Λ R ′ ,E ′ ⊙ B ′ , A ′ ) − → Hom R ′ (Λ R,E ⊙ B ′ , A ′ ) is a bijection. Since Ring R ′ is a full subca tegory of Ring R , the rig ht-hand side agre es with Hom R (Λ R,E ⊙ B ′ , A ′ ), and s o the map ab ov e is an isomor phism by (1.2 0.2). 1.21. T eichm¨ ul ler lifts. Let A b e an R -algebr a, let A ◦ denote the co mmutative monoid of all elements of A under m ultiplication, and let R [ A ◦ ] denote the monoid algebra on A ◦ . Then for each α ∈ E , the monoid endomorphism a 7→ a q α of A ◦ induces an R -algebra endomor phism ψ α of R [ A ◦ ] which reduces to the q α -th p ow er map mo dulo m α . Since R [ A ◦ ] is free a s an R - mo dule, it is ﬂa t. And since the v a rious ψ α commute with ea ch other, they provide R [ A ◦ ] with a Λ R,E -structure. Combined with the R -algebr a map R [ A ◦ ] → A g iven by the counit of the eviden t adjunction, this g ives, by the r ight-adjoin t prop erty of W R,E , a Λ R,E -ring map t : R [ A ◦ ] → W R,E ( A ). W e write the comp o s ite monoid map A ◦ unit − → R [ A ◦ ] ◦ t ◦ − → W R,E ( A ) ◦ 16 J. BOR GE R as simply a 7→ [ a ]. It is a sectio n of the R -alge br a map w 0 : W R,E ( A ) → A and is easily s e en to be functorial in A . The e le ment [ a ] is ca lled the T eichm¨ ul ler lift of a . 2. Grading and trunca tions 2.1. Or dering on Z ( E ) . F or t wo elements n ′ , n ∈ Z ( E ) = L E Z , write n ′ 6 n if w e hav e n ′ α 6 n α for all α ∈ E . Also put [0 , n ] = { n ′ ∈ N ( E ) | n ′ 6 n } . 2.2. T run c ations. W e have the following decomp ositio n of Ψ R,E : Ψ R,E = O α ∈ E O i ∈ N R [ ψ ◦ i α ] = O n ∈ N ( E ) R [ ψ n ] = R [ ψ n | n ∈ N ( E ) ] . (Th us, Ψ R,E is an N ( E ) -indexed copro duct in the categor y of R - algebra s , muc h like gra ded ring s a re monoid-indexed copro ducts in the category of modules. One might say that Ψ R,E is an N ( E ) -graded plethory . This p oint of view will not b e used b elow.) F or each n ∈ Z ( E ) , put Ψ R,E ,n = O α ∈ E O 0 6 i 6 n α R [ ψ ◦ i α ] = O n ′ ∈ [0 ,n ] R [ ψ n ′ ] = R  ψ n ′ | n ′ ∈ [0 , n ]  . Then Ψ R,E ,n represents the Ring R -v alued functor that sends A to the pr o duct ring A [0 ,n ] , which is natur ally a quotient of A N ( E ) . Deﬁne a s imilar ﬁltra tion on Λ R,E by (2.2.1) Λ R,E ,n = Λ R,E ∩  R [1 /E ] ⊗ R Ψ R,E ,n  . W e will often use the s hortened forms Λ E ,n , Ψ E ,n , Λ m ,n , Ψ m ,n , and so on. 2.3. Prop ositio n. (a) F or e ach n ∈ N ( E ) , the R -s cheme Spec Λ R,E ,n admits a unique structur e of an R -algebr a obje ct in the c ate gory of R -schemes such that the m ap Spec Λ R,E → Spec Λ R,E ,n induc e d by the inclusion Λ R,E ,n ⊆ Λ R,E is a homomorph ism of R -algebr a schemes over R . (b) F or e ach m, n ∈ N ( E ) , we have (2.3.1) Λ R,E ,m ◦ Λ R,E ,n ⊆ Λ R,E ,m + n , wher e ◦ denotes the c omp osition m ap of (1.18.1). Pr o of. (a): W rite Λ = Λ R,E , Λ n = Λ R,E ,n , and so o n. First observe that, for any int eger i > 0, a ll the ma ps in the diagr am R [1 /E ] ⊗ R Ψ ⊗ R i n a i / / R [1 /E ] ⊗ R Ψ ⊗ R i Λ ⊗ R i n b i / / O O Λ ⊗ R i c i O O are injective. Indeed, a i clearly is; the vertical maps are b ecause they b ecome isomorphisms after ba se change to R [1 /E ] and b eca use Λ n and Λ ar e E -ﬂat; and it follows formally that b i is injectiv e. Then the uniqueness of the desired R -alge br a scheme structur e on Spe c Λ n , follows from the injectivit y of b 2 . Now consider ex istence. Let ∆ : R [1 /E ] ⊗ R Ψ − → R [1 /E ] ⊗ R Ψ ⊗ R Ψ denote the ring map that induces the addition (res p. multiplication) map on the ring scheme Sp ec R [1 /E ] ⊗ R Ψ R . T o show tha t the desired addition and multiplication maps on Sp ec Λ n exist, it is enough to show (2.3.2) ∆(Λ n ) ⊆ Λ n ⊗ R Λ n . THE BASIC GEOMETR Y OF W ITT VECTORS, I 17 In fact, once w e do this, we will b e done: b ecause ea ch c i ◦ b i is injective, the ring axioms (asso cia tivity , distributivity , . . .) will follow fro m thos e on Sp ec R [1 /E ] ⊗ R Ψ. The map ∆ sends ψ α to ψ α ⊗ 1 + 1 ⊗ ψ α (resp. ψ α ⊗ ψ α ). Therefor e we have ∆( R [1 /E ] ⊗ R Ψ n ) ⊆ R [1 /E ] ⊗ R Ψ n ⊗ R Ψ n , and hence ∆(Λ n ) ⊆ Λ ⊗ R 2 ∩  R [1 /E ] ⊗ R Ψ ⊗ R 2 n  = Λ ⊗ R 2 n . This establishes (2.3.2) and hence completes the pro of of (a ). (b): Combine the deﬁnition (2.2 .1) with the inclusion ( R [1 /E ] ⊗ R Ψ m ) ◦ ( R [1 /E ] ⊗ R Ψ n ) ⊆ ( R [1 /E ] ⊗ R Ψ m + n ) and the inclus ion Λ m ◦ Λ n ⊆ Λ.  2.4. Witt ve ctors of ﬁ nite length. Let W R,E ,n denote the functor Ring R → Ring R represented by Λ R,E ,n : (2.4.1) W R,E ,n ( A ) = Hom R (Λ R,E ,n , A ) . W e call W R,E ,n the E -typic al Witt ve ctor functor of length n . As in 1.12, w e will often write W E ,n or W n ; when E = { m } , we will also wr ite W R, m ,n or W m ,n . W e then hav e (2.4.2) W R,E ( A ) = lim n W R,E ,n ( A ) . (Note that it is often better to view W R,E ( A ) as a pr o-ring than to actually take the limit. If we preferr ed top olog ical r ings to pro -rings, we could take the limit and endow it with the natura l pro- discrete top olog y .) It follows from 4.4 and (5.4.2) below that the maps in this pro jective system are surjective. The (truncated) ghost map (2.4.3) w 6 n : W R,E ,n ( A ) − → A [0 ,n ] , is the one induce d by the inclusion Ψ R,E ,n ⊆ Λ R,E ,n of repr esenting ob jects. F or any i ∈ [0 , n ], the comp osition w 6 n with the pro jection onto the i - th factor giv es another na tur al map (2.4.4) w i : W R,E ,n ( A ) − → A. Also the con ta inmen t (2.3.1) induces an R -algebra map (2.4.5) W R,E ,m + n ( A ) − → W R,E ,n  W R,E ,m ( A )  which sends an e lement a : Λ R,E ,m + n → A of W R,E ,m + n ( A ) to the map γ 7→ [ β 7→ a ( β ◦ γ )], for v aria bles γ ∈ Λ R,E ,n and β ∈ Λ R,E ,m . W e will call (2.4.5) c o-plethysm . It ag rees with the map of functor s induced b y the map (2.4.6) Λ R,E ,m ⊙ Λ R,E ,n − → Λ R,E ,m + n , β ⊙ γ 7→ β ◦ γ on repre s entin g ob jects, where β ⊙ γ is deﬁned as in [5]. Finally , observe that for any element f ∈ Λ R,E ,n the natural Λ R,E -ring op eration f : W R,E ( A ) → W R,E ( A ) (a map of sets) des cends to a map f : W R,E ,m + n ( A ) → W R,E ,m ( A ). Indeed, it is the comp os ition (2.4.7) W R,E ,m + n ( A ) ( 2 . 4 . 5 ) − → W R,E ,n  W R,E ,m ( A )  = Hom(Λ R,E ,n , W R,E ,m ( A )) − ( f ) − → W R,E ,m ( A ) , 18 J. BOR GE R where − ( f ) denotes the map that ev aluates at f . Particularly impor tant is the example f = ψ n , where the induced map (2.4.8) ψ n : W R,E ,m + n ( A ) → W R,E ,m ( A ) is a ring ho momorphism. 2.5. R emark: tr aditional versus normalize d indexing. Consider the p -typical Witt vectors, where R is Z and E consists of the s ingle ideal p Z . Let W ′ n denote Witt’s functor, as deﬁned in [26]. So, for example, W ′ n ( F p ) = Z /p n Z . In 3.5, we will construct a n is o morphism W ′ n +1 ∼ = W n . Thus, up to a no rmalization o f indices, o ur truncated Witt functors agr ee w ith Witt’s. The rea son for this nor malization is to make the indexing be hav e well under pleth ysm. By (2.3.1) and (2.4.5), the index s et ha s the structur e o f a commutativ e monoid, and so it is prefer able to use an index set with a familiar monoid structure. If we w er e to insist on agr eement with Witt’s indexing, w e would have to r eplace the sum m + n in (2.3.1) and (2.4.5) with m + n − (1 , 1 , . . . ), wher e this would b e computed in the pro duct gro up Z E . The reas on wh y this has not co me up in ea r lier work is tha t the pleth y sm structure has traditionally been used only thr o ugh the F r ob enius ma ps ψ α . In other words, only the s hift op e rator on the indexing set was used. Thus the distinction b etw een N and Z > 1 was not so important be cause the shift op er ator n 7→ n + 1 is written the same way on b oth. But making the ident iﬁcation of N and Z > 1 a monoid iso mo rphism would in volve the unw elcome addition law m + n − 1 on Z > 1 . It is diﬀerent with the big Witt vectors, where R is Z a nd E consists of all maximal ideals (1.15). They ar e also traditionally indexed by the positive integers ([16], (17.4.4)), but her e the po s itive integers are used m ultiplicatively rather than additively . In particular , the mono id structure that is re quired is the o bvious one; so the traditional indexing is in agreement with the normalized one: the big Witt ring W p n ( A ) (using traditional multiplicativ e indexing ) is naturally isomorphic to our p -typical ring W n ( A ) and to Witt’s W ′ n +1 ( A ). 2.6. L o c alization of the ring R of sc alars. Let R ′ be an E -ﬂat R -a lgebra such that the structur e map R → R ′ is an epimorphism of ring s , as in 1.20. Then for ea ch n ∈ N ( E ) , we hav e R ′ ⊗ R Λ R,E ,n = R ′ ⊗ R  Λ R,E ∩ ( R [1 /E ] ⊗ R Ψ R,E ,n )  ∼ →  R ′ ⊗ R Λ R,E  ∩ ( R ′ [1 /E ] ⊗ R ′ Ψ R ′ ,E ′ ,n ) . (W e only need to check that the displayed map is a n iso morphism along E , in which case it is is true b eca use R ′ is E -ﬂat over R .) By (1.20.1), this gives an isomor phis m of R ′ -algebra s (2.6.1) R ′ ⊗ R Λ R,E ,n ∼ − → Λ R ′ ,E ′ ,n . The induced isomorphism of represented functors gives natura l maps (2.6.2) W R ′ ,E ′ ,n ( A ′ ) ∼ − → W R,E ,n ( A ′ ) , for R ′ -algebra s A ′ . If A is an R -alge br a, the inv e rse of this map induces a map (2.6.3) R ′ ⊗ R W R,E ,n ( A ) − → W R ′ ,E ′ ,n ( R ′ ⊗ R A ) W e will see in 6.1 that this is an isomorphism. As with (1.2 0.3), the map (2.6.2) induces an isomor phism (2.6.4) Λ R,E ,n ⊙ B ′ ∼ − → Λ R ′ ,E ′ ,n ⊙ B ′ , for any R ′ -algebra B ′ , THE BASIC GEOMETR Y OF W ITT VECTORS, I 19 2.7. Prop o sition. L et A b e an E -ﬂat R -algebr a. Then the ghost map w 6 n : W R,E ,n ( A ) − → A [0 ,n ] is inje ct ive. If A is an R [1 / E ] -algebr a, it is an isomorphism. Recall that the analo gous facts for inﬁnite-length Witt vectors are a lso true, either by construction (1.8 ) or by the universal prop erty (1.9). Pr o of. If every ideal in E is the unit ideal, then Λ R,E = Ψ R,E , and hence w e hav e Λ R,E ,n = Ψ R,E ,n . The s ta tement ab out R [1 /E ]-alg ebras then follows from (2.6.1) . The statement about E -ﬂat R -alg e bras follo ws b y co nsidering the injection A → R [1 /E ] ⊗ R A and a pplying the previous case to R [1 /E ] ⊗ R A .  3. Princip al single-prime case F o r this section, we will restrict to the case where E consists of one ideal m generated by an element π . Our purpo se is to extend the classical comp onents of Witt vectors fr om the p -typical co nt ext (where R is Z and E consists of the single ideal p Z ) to this slightly more general one. The reaso n for this is that the Witt comp onents ar e w ell- suited to calculation. In the following sections, we will see how to use them, together with 4.1, 5.4, a nd 6.1, to draw co nclusions when E is ge neral. In fact, the usual a r guments and deﬁnitions in the cla ssical theory o f Witt vectors carry over a s long as one mo diﬁes the usual Witt p olynomials by replacing every p in an e x po nent with q m , and e very p in a co eﬃcient with π . Some things, suc h as the V er schiebung op er ator, dep end o n the choice of π , and others do not, suc h as the V erschiebung ﬁltration. Let n denote an element of N . Let us a bbreviate Λ m = Λ R,E , Λ m ,n = Λ R,E ,n , W m = W R,E , q = q m , ψ = ψ m , and so on. 3.1. θ op er ators. Deﬁne elemen ts θ π , 0 , θ π , 1 , . . . of R [1 /π ] ⊗ R Λ m = R [1 / π ] ⊗ R Ψ m recursively by the g eneralized Witt polyno mials (3.1.1) ψ ◦ n = θ q n π , 0 + π θ q n − 1 π , 1 + · · · + π n θ π ,n . (Note that the exp o nent on the left side means itera ted comp osition, while the exp onents on the rig ht mea n usual exp onentiation, iter ated multiplication.) As in 1.5, w e ca n view the elements θ π ,i as natura l op er ators on Ψ R [1 /π ] , m -rings. W e will often write θ i = θ π ,i when π is c lear. 3.2. Lemm a. We have (3.2.1) ψ ◦ θ π ,n = θ q π ,n + π θ π ,n +1 + π P ( θ π , 0 , . . . , θ π ,n − 1 ) , for some p olynomial P ( θ π , 0 , . . . , θ π ,n − 1 ) with c o eﬃcients in R . Pr o of. It is clea r for n = 0 . F o r n > 1, we will use induction. Recall the gener al implication x ≡ y mo d m = ⇒ x q j ≡ y q j mo d m j +1 , for j > 1, which itself is e a sily pr ov ed by induction. T oge ther with the formula (3.2.1) for ψ ◦ θ π ,i with i < n , this implies ψ ◦ ψ ◦ n = n X i =0 π i ( ψ ◦ θ i ) q n − i ≡ π n ψ ◦ θ n + n − 1 X i =0 π i ( θ q i ) q n − i mo d m n +1 R [ θ 0 , . . . , θ n − 1 ] 20 J. BOR GE R When this is combined with the deﬁning fo rmula (3.1 .1) for ψ ◦ ( n +1) , we hav e π n ψ ◦ θ n ≡ π n θ q n + π n +1 θ n +1 mo d m n +1 R [ θ 0 , . . . , θ n − 1 ] . Dividing by π n completes the pro of.  3.3. Prop o sition. The elements θ π , 0 , θ π , 1 , . . . of R [1 / π ] ⊗ R Λ m lie in Λ m , and they gener ate Λ m fr e ely as an R -algebr a. F urther, the element s θ π , 0 , . . . , θ π ,n lie in Λ m ,n , and they gener ate Λ m ,n fr e ely as an R - algebr a. This is essent ially Witt’s theorem 1 [26]. Pr o of. By induction, the elements θ 0 , . . . , θ n generate the s ame sub- R [1 /π ]- algebra of R [1 /π ] ⊗ R Λ m as ψ ◦ 0 , . . . , ψ ◦ n , and are hence algebra ically indep endent ov e r R [1 /π ]. Since R ⊆ R [1 /π ], they are also algebraically indep endent ov er R . Let B n be the sub- R -algebr a o f R [1 /π ] ⊗ R Λ m generated by θ 0 , . . . , θ n , and let B = S n B n . T o show Λ m ⊇ B , we may a ssume by induction that Λ m ⊇ B n and then show Λ m ⊇ B n +1 . By 3.2 and b ecause Λ m is a Λ m -ring, we hav e π θ n +1 ∈  ψ ◦ θ n − θ q n  + m Λ m ,n ⊆ m Λ m . Dividing by π , w e have θ n +1 ∈ Λ m , and hence Λ m ⊇ B n [ θ n +1 ] = B n +1 . On the o ther hand, b y 3.2 aga in, we hav e ψ ◦ θ n ≡ θ q n mo d m B n +1 for all n . Ther efore B , sinc e it is genera ted by the θ n , is a sub-Λ m -ring of R [1 /π ] ⊗ R Λ m . It follows that B ⊇ Λ m ◦ e = Λ m and hence that B = Λ m . Last, the equalit y Λ m ,n = B n follows immediately from the above: Λ m ,n = Λ m ∩  R [1 /π ] ⊗ R Ψ m ,n  = B ∩  R [1 /π ] ⊗ R Ψ m ,n  = R [ θ 0 , . . . ] ∩ R [1 /π ][ θ 0 , . . . , θ n ] = R [ θ 0 , . . . , θ n ] = B n .  3.4. Example: Pr esentations of Λ m ,n ⊙ A . Using 3.3, w e can turn a prese ntation of an R -algebr a A into a pre s entation of Λ m ,n ⊙ A . W e have Λ m ,n ⊙ R [ x ] ∼ = Λ m ,n = R [ θ 0 , . . . , θ n ] , where θ k is sho rt for θ π ,k , which corr esp onds to the element θ π ,k ( x ) = θ π ,k ⊙ x . Because the functor Λ m ,n ⊙ − preserves copr o ducts and co equalizers, we hav e (3.4.1) Λ m ,n ⊙  R [ x 1 , . . . , x r ] / ( f 1 , . . . , f s )  = R [ θ i ( x j )] / ( θ i ( f k )) , where 0 6 i 6 n , 1 6 j 6 r , and 1 6 k 6 s . Here each expr ession θ i ( x j ) is a sing le free v aria ble, and θ i ( f k ) is under sto o d to b e the p olyno mia l in the v a riables θ i ( x j ) that r esults from expanding θ i ( f k ) using the s um a nd pro duct laws for θ i . Because Λ m ,n ⊙ − pr e serves ﬁltered colimits, we can give a similar pr esentation of Λ m ,n ⊙ A for any R - algebra A . Simila rly , we can take the colimit ov er n to g et a pres e ntation for Λ m ⊙ A . In the E -t ypica l case, where E is ﬁnite, one can wr ite down a presentation of Λ R,E ⊙ A by iterating (3.4.1), acco rding to 5.3 b elow. W e can pa ss from the ca se where E is ﬁnite to the case where it is arbitra ry by taking colimits, as in 5.1. The metho d above is not particular to the θ o p erators —it works for any subset of Λ m ,n that gener a tes it freely a s an R -algebr a. F or exa mple, w e c an use the δ op erator s of 1 .1 9. Let δ i ∈ Λ m denote the i -th itera te of δ π . Then the elements δ 0 , . . . , δ n lie in Λ m ,n and freely genera te it as a n R -algebr a. (As in 3 .3, this follows THE BASIC GEOMETR Y OF W ITT VECTORS, I 21 by induction, but in this cas e, there are no subtle congr uences to check.) Therefore we hav e (3.4.2) Λ m ,n ⊙  R [ x 1 , . . . , x r ] / ( f 1 , . . . , f s )  = R [ δ i ( x j )] / ( δ i ( f k )) , where 0 6 i 6 n , 1 6 j 6 r , a nd 1 6 k 6 s . W e in terpr et the expre s sions δ i ( x j ) and δ i ( f k ) a s ab ov e. The g eneral E -typical case can b e handled a s ab ov e . (See Buium–Simanca [9], pro of of propo sition 2.12.) 3.5. Witt c omp onents. It follows from 3 .3 that, given π , w e ha ve a bijection (3.5.1) W m ( A ) ∼ − → A × A × · · · , which sends a map f : Λ m → A to the sequence ( f ( θ π , 0 ) , f ( θ π , 1 ) , . . . ). T o make the dependence on π explicit, we will often write ( x 0 , x 1 , . . . ) π for the image of ( x 0 , x 1 , . . . ) under the inv er se of this map. If R = Z and π = p , then this iden tiﬁes W m ( A ) with the ring of p -t y pica l Witt vectors a s deﬁned traditionally . Similarly , when R is a co mplete discrete v alua tion ring , we get a n identiﬁcation of W m ( A ) with Hazewinkel’s ring of ra miﬁed Witt vectors W R q, ∞ ( A ). (See [16], (18 .6.13), (25.3 .17), and (25.3 .26)(i).) W e call the x i the Witt c omp onents (rela tive to π ) of the ele ment ( x 0 , . . . ) π ∈ W ( A ). Similarly , using the free gener ating s e t θ π , 0 , . . . , θ π ,n of Λ m ,n , we hav e a bijection (3.5.2) W m ,n ( A ) ∼ − → A [0 ,n ] . As ab ov e, we will wr ite ( x 0 , . . . , x n ) π for the image o f ( x 0 , . . . , x n ) under the inv erse of this map. This identiﬁ es W m ,n ( A ) with the traditionally deﬁned ring of p -typical Witt vectors o f length n + 1. (F or remarks on the +1 shift, see 2.5.) Note that the Witt comp onents do not dep end on the choice of π in a simple, m ultilinear wa y . F or example, if u is an inv ertible e le ment of R and we hav e ( x 0 , x 1 , . . . ) π = ( y 0 , y 1 , . . . ) uπ , then we have x 0 = y 0 , x 1 = uy 1 , x 2 = u 2 y 2 + π − 1 ( u − u q ) y q 1 , . . . . As in 3.4, we co uld use the free genera ting set δ 0 , δ 1 , . . . of Λ m instead of θ 0 , θ 1 , . . . . This would give a diﬀerent bijection b etw een W m ( A ) and the set A N , a nd hence a n R -alg ebra structure on the set A N which is isomorphic to Witt’s but not equal to it. The truncated versions ag ree up to A × A , but diﬀer after that. This is simply bec ause δ 0 = θ 0 and δ 1 = θ 1 , but δ 2 6 = θ 2 . (See Joy al [19], p. 179.) 3.6. The ghost principl e. It follows from the descriptions (3.5.1) a nd (3.5.2) that W m and W m ,n preserve sur jectivity . On the o ther hand, every R - algebra is a quo- tien t of an m -ﬂa t R -alg ebra (ev en a free one). Ther efore to prove any functor ial ident it y inv olving rings of Witt vectors when m is pr inc ipa l, it is enough to restrict to the m - ﬂat case. F urther, any m - ﬂat R - algebra A is contained in an R [1 / m ]-alg ebra, such a s R [1 / m ] ⊗ R A . Since W m and W m ,n , b eing repre sentable functors, preserve injectivit y , it is even enough to chec k functorial iden tities o n R [1 / m ]-alg ebras A , in which ca se ring s of Witt vectors agr ee with the muc h mor e tracta ble r ings o f ghos t comp onents. An example with details is given in 3 .7. 3.7. V erschiebung. F or any R - algebra A deﬁne an op erato r V π , c a lled the V er- schiebung (rela tive to π ), on W m ( A ) by (3.7.1) V π  ( y 0 , y 1 , . . . ) π  = (0 , y 0 , y 1 , . . . ) π . This is clea r ly functoria l in A . Deﬁne another, identically deno ted op erato r on the ghost ring A N by the form ula (3.7.2) V π  h z 0 , z 1 , . . . i  = h 0 , π z 0 , π z 1 , . . . i . 22 J. BOR GE R These o pe r ators are compa tible in that w e have w ( V π ( y )) = V π ( w ( y )) for all y ∈ W m ( A ), a nd the op erator V π on the g ho st ring is clea r ly R -linea r. It follows by the ghost principle tha t the op era tor V π on W m ( A ) is R -linear . Here is the arg ument in so me detail. W e need to chec k the identities r V π ( y ) = V π ( ry ) and V π ( x + y ) = V π ( x ) + V π ( y ), for r ∈ R , x, y ∈ W m ( A ). W rite x = ( x 0 , x 1 , . . . ) π and y = ( y 0 , y 1 , . . . ) π . If A is a E -ﬂat, the g host ma p w : W m ( A ) → A N is injectiv e. Therefor e V π is R -linear o n W m ( A ), by the R -linearity of V π on the g host ring . The ge ne r al cas e then follows from E -ﬂat cas e . Fix an E -ﬂat R -algebra ˜ A with a surjective R -algebr a map ˜ A → A . F or each i , let ˜ y i be a pre-imag e of y i , a nd set ˜ y = ( ˜ y 0 , . . . ) π ∈ W m ( A ). The induced map f : W m ( ˜ A ) → W m ( A ) then satisﬁes f ( ˜ y ) = y . Therefo r e we have V π ( ry ) = V π ( rf ( ˜ y )) = V π ( f ( r ˜ y )) = f ( V π ( r ˜ y )) = f ( r V π ( ˜ y )) = rf ( V π ( ˜ y )) = rV π ( f ( ˜ y )) = r V π ( y ) . The additivity axiom follows similarly . 3.8. Example. W R, m ,n ( R ) ha s a pr esentation R [ x 1 , . . . , x n ] / ( x i x j − π i x j | 1 6 i 6 j 6 n ) , where the elemen t x i corres p o nds to V i π (1). 3.9. T eichm¨ ul ler lifts. Under the comp osition A a 7→ [ a ] / / W ( A ) w / / A × A × · · · (see 1.21), the image of a is h a, a q , a q 2 , . . . i . It follows from the gho s t principle that [ a ] = ( a, 0 , 0 , . . . ) π ∈ W ( A ) . Multiplication b y T eichm¨ uller lifts also has a simple description in terms of Witt comp onents: (3.9.1) [ a ]( . . . , b i , . . . ) π = ( . . . , a q i b i , . . . ) π . Again, this follo ws from the ghost principle. 4. General single-prime case Assume E cons ists of a single ide a l m , p oss ibly not principal. Let n be an e lement of N . Let us wr ite W R, m ,n = W R,E ,n and so on. Let K m denote R m [1 / m ]. If m is the unit ideal, we understand R m , and hence K m , to b e the zero ring. Otherwise, R m is a discrete v alua tio n ring and K m is its fraction ﬁeld. In particular, m b eco mes principal in R [1 / m ], R m , and K m . The following prop o sition then allows us to describ e W R, m ,n ( A ) in terms o f the case where m is principal, a nd hence in terms of Witt comp onents. 4.1. Prop ositio n. F or R ′ = R [1 / m ] , R m , K m , write W R ′ , m ,n = W R ′ , m R ′ ,n . Then for any R -algebr a A , the ring W R, m ,n ( A ) is the e qualizer of the two maps W R [1 / m ] , m , n  R [1 / m ] ⊗ R A  × W R m , m ,n  R m ⊗ R A  / / / / W K m , m ,n ( K m ⊗ R A ) induc e d by pr oje ction onto the two factors and the bifunctoriality of W − , m ,n ( − ) . Pr o of. The diagram R / / R [1 / m ] × R m pr 1 / / pr 2 / / K m THE BASIC GEOMETR Y OF W ITT VECTORS, I 23 is an equalizer diagr am. Since K m is m -ﬂat, so is any sub- R -mo dule of K m . It follows that for any R -algebr a A , the induced diag ram A / /  R [1 / m ] × R m  ⊗ R A / / / / K m ⊗ R A is a n eq ualizer dia gram. Since W R, m ,n is r epresentable, it preser ves equalizer s , and so the induced diagra m (wr iting W n = W R, m ,n ) W n ( A ) / / W n  R [1 / m ] ⊗ R A  × W n  R m ⊗ R A  / / / / W n ( K m ⊗ R A ) is also an equalizer dia gram. Then (2.6.2) completes the pro of.  4.2. V erschiebung in gener al. W e can deﬁne V erschiebung maps (4.2.1) V j : m j ⊗ R W R, m ( A ) − → W R, m ( A ) . T o do this, it is enough, by 4.1, to restrict to the c a se wher e m is principal, as lo ng as our constructio n is functorial in A a nd R . So, choos e a generator π ∈ m and deﬁne (4.2.2) V j ( π j ⊗ y ) = V j π ( y ) , for all y ∈ W R, m ( A ). On ghost comp onents it satisﬁes V j ( x ⊗ h z 0 , z 1 , . . . i ) = h 0 , . . . , 0 , xz 0 , xz 1 , . . . i , where the num b er of leading zeros is j . In particula r, it is indepe ndent of the choice of π , by the ghost principle. If we wr ite W R, m ( A ) ( j ) for W R, m ( A ) viewed as a W R, m ( A )-algebra by wa y of the map ψ j : W R, m ( A ) → W R, m ( A ), then the map (4.2.3) V j : m j ⊗ R W R, m ( A ) ( j ) − → W R, m ( A ) , is W R, m ( A )-linear, as is easily check ed using the gho st principle. Expr essed as a formula, it says (4.2.4) V j ( x ⊗ y ψ j ( z )) = V j ( x ⊗ y ) z . In par ticular, the imag e V j W R, m ( A ) of V j is an ideal of W R, m ( A ). Let us a lso recor d the iden tities (4.2.5) ψ j  V j ( x ⊗ y )  = xy and (4.2.6) V j ( x ⊗ y ) V j ( x ′ ⊗ y ′ ) = xV j ( x ′ ⊗ y y ′ ) ∈ m j V j W R, m ( A ) . Again, one checks these using the gho st principle. Finally , for a ny n ∈ N , the map V j descends to a map (4.2.7) V j : m j ⊗ R W R, m ,n ( A ) ( j ) − → W R, m ,n + j ( A ) , and the o bvious a nalogues of the iden tities ab ove hold here. 4.3. R emark. W e can deﬁne V erschiebung maps even if we no lo ng er assume there is only one ideal in E . F o r any j ∈ N ( E ) , let J denote the ideal Q α m j α α of R . Then V j would be a map J ⊗ R W R,E ( A ) → W R,E ( A ). The iden tities ab ov e, suitably int erpreted, co ntin ue to hold. W e will not need this multiple-prime version. 4.4. Prop ositio n. The se quen c e (4.4.1) 0 − → m j ⊗ R W R, m ,n ( A ) ( j ) V j − → W R, m ,n + j ( A ) − → W R, m ,j ( A ) − → 0 is exact. 24 J. BOR GE R Pr o of. W rite W R ′ ,n = W R ′ , m R ′ ,n when R ′ is a n R -algebra s uch that the ideal m R ′ is suprama ximal. First consider the case where m is principa l. Let π ∈ m b e a gener ator. Us- ing (3.7.1), it is clear that V j is injective and that its image is the set of Witt vectors w ho se Witt components (relative π ) ar e 0 in p ositions 0 to j − 1 . By 3.5, the pre-image of 0 under the map W R,n + j ( A ) → W R,j ( A ) is the same subset, and the map W R,n + j ( A ) → W R,j ( A ) is surjective. Now consider the genera l ca se. Augment the diagram (4.4.1) by ex pr essing each term of (4.4.1) a s an equalizer as in 4.1. Here we use that m is R - ﬂa t. It then follows fro m the pr incipal case and the snake lemma that (4.4.1) is left exact. It remains to prove that the map W R,n + j ( A ) → W R,j ( A ) is surjective. By induction, we can ass ume n = 1. By 4 .1, for any i ∈ N we have W R,i ( A ) = W R m ,i ( R m ⊗ R A ) × W K m ,i ( K m ⊗ R A ) W R [1 / m ] ,i ( R [1 / m ] ⊗ R A ) . Now let π denote a genera tor of the maximal ideal of R m , and suppo se tw o elements y = ( y 0 , . . . , y j ) π ∈ W R m ,j ( R m ⊗ R A ) , z = h z 0 , . . . , z j i ∈ ( R [1 / m ] ⊗ R A ) j +1 = W R [1 / m ] ,j ( R [1 / m ] ⊗ R A ) hav e the same ima ge in W K m ,j ( K m ⊗ R A ). T o lift the corr esp onding element of W j ( A ) to W j +1 ( A ), we need to ﬁnd elements y j +1 ∈ R m ⊗ R A and z j +1 ∈ R [1 / m ] ⊗ R A such that in K m ⊗ R A , we have (4.4.2) y q j +1 0 + · · · + π j +1 y j +1 = z j +1 . So, choos e an element z j +1 ∈ A whose image under the surjection A − → A/ ( m A ) j +1 = R m / ( m R m ) j +1 ⊗ R A agrees with the image of y q j +1 0 + · · · + π j y j . It follows that the element y q j +1 0 + · · · + π j y q j − 1 ⊗ z j +1 ∈ R m ⊗ R A lies in π j +1 ( R m ⊗ R A ). It thus equals π j +1 y j +1 for some element y j +1 ∈ R m ⊗ R A . And so y j +1 and z j +1 satisfy (4.4.2).  4.5. Corollary . F or any R -algebr a A , we have (4.5.1) M i ∈ [0 ,n ] m i ⊗ R A ( i ) ∼ − → gr V W R, m ,n ( A ) , wher e A ( i ) denotes A viewe d as a W n ( A ) -mo dule via the ring map w i : W n ( A ) → A . 4.6. R e duc e d ghost c omp onents. W e can deﬁne inﬁnitely man y gho st compo nent s for Witt vectors of ﬁnite length n if we are willing to settle for answers modulo m n +1 . First ass ume m is gene r ated by so me element π . By examining the Witt p olyno- mials (3.1.1), w e can see tha t for an y i > 0, the comp osition W R, m ( A ) w i − → A − → A/ m n +1 A v a nishes on V n +1 W R, m ( A ). It ther efore factors through W R, m ,n ( A ), giving a map ¯ w i from W R, m ,n ( A ) to A/ m n +1 A . When m is not ass umed to b e pr incipal, we de ﬁne ¯ w i by lo calizing at m : W R, m ,n ( A ) − → W R m , m R m ,n ( R m ⊗ R A ) ¯ w i − → ( R m ⊗ R A ) / m n +1 ( R m ⊗ R A ) = A/ m n +1 A, THE BASIC GEOMETR Y OF W ITT VECTORS, I 25 where the middle map is ¯ w i as constructed ab ov e in the principal ca se. W e call the comp osition (4.6.1) W R, m ,n ( A ) ¯ w i − → A/ m n +1 A the i -th r e duc e d ghost c omp onent map. 5. Mul tiple-prime case The purp os e of this section is to give some results on reducing the family E (of 1.2) to simpler families. The ﬁrst reduces from the case where E is ar bitrary to the case where it is ﬁnite, and the se c ond reduces from the case where it is ﬁnite to the ca se where it has a single element. W e will often write W E = W R,E , Λ E = Λ R,E , and so o n, for short. 5.1. Prop ositio n. The c anonic al maps colim E ′ Λ R,E ′ − → Λ R,E , (5.1.1) colim E ′ Λ R,E ′ ,n ′ − → Λ R,E ,n , (5.1.2) ar e isomorphisms. Her e E ′ runs over the ﬁnite su bfamilies of E , and n ′ is the r estriction to E ′ of a given element n ∈ N ( E ) . Pr o of. Cons ider (5.1.1) ﬁrst. Since ea ch map Λ E ′ → Λ E is an injection, (5.1.1) is an injection. Therefore, since Λ E is freely generated as a Λ E -ring by the e le ment e = ψ 0 , it is eno ugh to show the sub-Ψ E -ring colim E ′ Λ E ′ of Λ E is a sub-Λ E -ring. Since it is ﬂat, we only nee d to check the F rob enius lift prop erty . So, s uppo se m ∈ E . F o r any e le men t x of the colimit, there is a ﬁnite family E ′′ such that x ∈ Λ E ′′ and m ∈ E ′′ . But Λ E ′′ is a Λ E ′′ -ring. So we hav e ψ m ( x ) ≡ x q m mo dulo m Λ E ′′ , and hence modulo m (co lim E ′ Λ E ′ ). There fo re the F rob enius lift prop erty holds for the colimit r ing. Then (5.1.2) follows: Λ E ,n = ( R [1 /E ] ⊗ R Ψ E ,n ) ∩ Λ E = (colim E ′ R [1 /E ] ⊗ R Ψ E ′ ,n ′ ) ∩ co lim E ′ Λ E ′ = colim E ′  ( R [1 /E ] ⊗ R Ψ E ′ ,n ′ ) ∩ Λ E ′  = colim E ′ Λ E ′ ,n ′ .  5.2. Corollary . F or any R -algebr a A , the c anonic al maps W R,E ( A ) − → lim E ′ W R,E ′ ( A ) , (5.2.1) W R,E ,n ( A ) − → lim E ′ W R,E ′ ,n ′ ( A ) (5.2.2) ar e isomorphisms, wher e E ′ , n , and n ′ ar e as in 5.1. 5.3. Prop ositio n. L et E ′ ∐ E ′′ b e a p artition of E . Then the c anonic al maps Λ R,E ′ ⊙ R Λ R,E ′′ − → Λ R,E (5.3.1) Λ R,E ′ ,n ′ ⊙ R Λ R,E ′′ ,n ′′ − → Λ R,E ,n (5.3.2) ar e isomorphisms, wher e n ′ and n ′′ denote the r estr ictions to E ′ and E ′′ of a given element n ∈ N ( E ) . Pr o of. It is enough to show each map b ecomes a n isomorphis m after ba se change to R [1 /E ′ ] and R [1 /E ′′ ]. So, by (1.20.1), we ca n assume every element in either E ′ or E ′′ is the unit ideal. In the sec ond case, w e have Λ E ′ ⊙ R Λ E ′′ = Λ E ′ ⊙ R R [ N ( E ′′ ) ] = Λ E ′ [ N ( E ′′ ) ] = Λ E 26 J. BOR GE R The a rgument for (5.3.2) is the sa me, but we r eplace the genera ting set N ( E ′′ ) with [0 , n ′′ ]. Now s uppo se every element in E ′ is the unit ideal. Then a Λ E ′ -ring is the same as a Ψ E ′ -ring. So w e hav e Λ E ′ ⊙ R Λ E ′′ = Λ E ′′ [ N ( E ′ ) ] = Λ E . F o r (5.3.2), replace N ( E ′ ) with [0 , n ′ ], as above.  5.4. Corollary . L et E ′ ∐ E ′′ b e a p art ition of E . Then for any R -algebr a A , the c anonic al maps W R,E ( A ) − → W R,E ′′  W R,E ′ ( A )  (5.4.1) W R,E ,n ( A ) − → W R,E ′′ ,n ′′  W R,E ′ ,n ′ ( A )  (5.4.2) ar e isomorphisms, wher e n , n ′ , n ′′ ar e as in 5.3. 5.5. R emark. By the res ults above, it is safe to say that expr essions such as (5.5.1) Λ m 1 ⊙ R · · · ⊙ R Λ m r and W m r ◦ · · · ◦ W m 1 ( A ) are independent o f the orde r ing of the m i , assuming the m i are pairwise co prime. (Note that it is not generally true that P ⊙ P ′ ∼ = P ′ ⊙ P for plethories P and P ′ . See [5], 2.8.) If we as k tha t the expres sions in (5.5 .1) be indep endent only up to isomorphis m, then it is no t even necessa ry that the m α ∈ E be pa irwise co pr ime (1.2). But inv ariance up to isomo rphism is not a such a useful prop erty , and mo st o f the time coprimality really is necessar y . F or example, we could lo ok at rings with mo re than one F rob enius lift at a single maximal idea l, but we would no t b e able to r educe to the case of a single F r ob enius lift. Indeed, if E consists of a single maximal ideal m , the tw o endomor phisms ψ W W ( A ) and W ( ψ W ( A ) ) of W W ( A ) co mm ute, and the ﬁrst is clearly a F rob enius lift, but the second is generally no t. Therefor e W W ( A ) cannot b e the cofree r ing with t wo commuting F rob enius lifts at m . In fact, I b elieve this is the only place where we use the co primality a s sumption directly . The re s t o f our r esults dep end o n it only through 5.3. Although I k now of no applications, it would b e interesting to know whether the a bstract set up of this pap er, and then the main r esults, hold when we allow mor e than o ne F rob enius lift at each ma ximal ideal. 6. Basic affine proper ties This section provides some basic r esults ab out the co mm utative alg e bra of Witt vectors. They are just the ones needed to b e able to prov e the main theo rems in sections 8 and 9 and to set up the glo bal theory in [4]. Ther e ar e other bas ic results that could have b een included here, but which I hav e put oﬀ to [4], where they will be prov ed for all algebra ic s paces. W e con tinue with the notation of 1.2. Fix an ele ment n ∈ N ( E ) . W e will o ften write W n = W E ,n = W R,E ,n and so on, for short. By 5.2, w e may a ssume that E agrees with the supp ort o f n , a nd in particular that it is ﬁnite. 6.1. Prop ositio n. L et R ′ b e an E -ﬂat R -algebr a such that the st ructur e map R → R ′ is a ring epimorph ism ( as in 1.20). Then the c omp osition R ′ ⊗ R W R,E ,n ( A ) ( 2 . 6 . 3 ) / / W R,E ,n ( R ′ ⊗ R A ) ∼ ( 2 . 6 . 2 ) − 1 / / W R ′ ,E ′ ,n ( R ′ ⊗ R A ) is an isomorphi sm, wher e E ′ is as in 1.20. THE BASIC GEOMETR Y OF W ITT VECTORS, I 27 Pr o of. W e may assume b y 5.4 that E consists of a single ideal m . Using 4.1 a nd the ﬂatness of R ′ ov er R , we ar e reduced to showing that the functors W R [1 / m ] , m , n , W R m , m ,n , and W K m , m ,n commute with the functor R ′ ⊗ R − . Therefo re we may assume that the ideal m is principal. W r ite W n = W R, m ,n . The r esult is clear for n = 0, b ecause W 0 is the identit y functor. So assume n > 1 . By 4.4, we have the follo wing map o f exact sequences 0 / / R ′ ⊗ R m ⊗ R W n − 1 ( A ) id R ′ ⊗ V 1 / /   R ′ ⊗ R W n ( A ) / /   R ′ ⊗ R A / / 0 0 / / m ⊗ R W n − 1 ( R ′ ⊗ R A ) V 1 / / W n ( R ′ ⊗ R A ) / / R ′ ⊗ R A / / 0 , where the vertical maps are g iven b y (2.6.3). By induction the leftmost vertical arrow is an isomorphism. T her efore the inner one is, too.  6.2. Prop ositio n. F or any ide al I in an R -algebr a A , let W R,E ,n ( I ) denote the kernel of the c anonic al map W R,E ,n ( A ) → W R,E ,n ( A/I ) . Then we have W R,E ,n ( I ) W R,E ,n ( J ) ⊆ W R,E ,n ( I J ) for any ide als I , J in A . Pr o of. Let us ﬁrst show that we may as s ume E co nsists o f a sing le ideal m . In doing this, it will b e conv enient to prov e the following equiv alent fo r m of the statemen t: if I J ⊆ K , where K is an ideal in A , then W n ( I ) W n ( J ) ⊆ W n ( K ). Supp ose E = E ′ ∐ { m } . Le t n ′ be the r estriction of n to E . Let I ′ = W E ′ ,n ′ ( I ), J ′ = W E ′ ,n ′ ( J ), and K ′ = W E ′ ,n ′ ( K ). By 5.4, we hav e W E ,n = W m ,n m ◦ W E ′ ,n ′ , a nd hence W E ,n ( I ) = W m ,n m ( I ′ ) and so on. By induction, we have I ′ J ′ ⊆ K ′ , and then applying the result in the single-ideal case gives W E ,n ( I ) W E ,n ( J ) = W m ,n m ( I ′ ) W m ,n m ( J ′ ) = W m ,n m ( K ′ ) = W E ,n ( K ) . So we will assume E = { m } and drop E from the notation. By 6.1, the statement is Zar iski lo cal on R , and so we may a ssume the ideal m is g enerated by s ome element π . W e will work with Witt comp onents re la tive to π . W e need to s how that for any elements x = ( x 0 , . . . , x n ) π ∈ W n ( I ) a nd y = ( y 0 , . . . , y n ) π ∈ W n ( J ), the pro duct xy is in W n ( I J ). So it is suﬃcient to show this in the universal case, where A is the free polyno mial algebr a R [ x 0 , y 0 , . . . , x n , y n ], I is the ideal ( x 0 , . . . , x n ), and J is the idea l ( y 0 , . . . , y n ). Consider the fo llowing co mm utative diagr am W n ( A ) w 6 n / /   A [0 ,n ]   W n ( A/I J ) w 6 n / / ( A/I J ) [0 ,n ] . W e wan t to show that the image of xy in W n ( A/I J ) is ze ro. Since A/I J is ﬂat (even free) over R , the lo wer map w 6 n is injective, and so it is e no ugh to show the image of xy in ( A/I J ) [0 ,n ] is zero. But by the na tur ality of the g host map, we have w 6 n ( x ) ∈ I [0 ,n ] and w 6 n ( y ) ∈ J [0 ,n ] . Therefor e w 6 n ( xy ) lies in ( I J ) [0 ,n ] , which maps to zero in ( A/I J ) [0 ,n ] .  6.3. R emark. Although the proo f of 6.2 given ab ov e uses s ome pr op erties speciﬁc to Witt vector functors, the r esult is tr ue for any r epresentable ring -v alued functor. See Bo rger– Wieland [5], 5 .5. 28 J. BOR GE R 6.4. Corollary . L et I b e an ide al in an R -algebr a A . If I m = 0 , then W R,E ,n ( I ) m = 0 . 6.5. Prop ositio n. L et ϕ : A → B b e a map of R -algebr as. If it is surje ctive, then so is the map W R,E ,n ( ϕ ) : W R,E ,n ( A ) → W R,E ,n ( B ) . Pr o of. By 5.4, we may a ssume E consists of o ne ideal m . Since sur jectivity can b e chec ked Zarisk i lo cally on R , it is enough by 6.1 to assume m is principa l. Then using the Witt comp onents, we can ident ify the set map underly ing W R,E ,n ( ϕ ) w ith the map ϕ [0 ,n ] : A [0 ,n ] → B [0 ,n ] , which is clea rly surjective.  6.6. Corollary . If ϕ : A → B is surje ctive, t hen W R,E ,n ( A × B A ) W n (pr 1 ) / / W n (pr 2 ) / / W R,E ,n ( A ) W n ( ϕ ) / / W R,E ,n ( B ) is a c o e qualizer diagr am. Pr o of. The functor W n is r epresentable, a nd hence commutes with limits. (See 2.4.) Therefore W n ( A × B A ) ag rees with W n ( A ) × W n ( B ) W n ( A ), which is a n equiv alence relation on W n ( A ), the quotient by which is the image of W n ( ϕ ). By 6.5, this is a ll of W n ( B ).  6.7. R emark. This result is particularly appea ling when A is E -ﬂat a nd B is not. Then we can describ e W n ( B ) in terms of W n ( A ) and W n ( A × B A ), w hich a re directly acces sible b ecause A and A × B A ar e E - ﬂat. 6.8. Prop ositio n. Supp ose E c onsists of one ide al m , and let A b e an R -algebr a. F or any i > 0 , the m ap Sp ec(id ⊗ ¯ w i ) of schemes induc e d by the ring map id ⊗ ¯ w i : R/ m ⊗ R W R,E ,n ( A ) − → R/ m ⊗ R A/ m n +1 is a un iversal home omorphism. F or i = 0 , it is a close d immersion deﬁne d by a squar e-zer o ide al. Pr o of. W rite W n = W R,E ,n and so on. Consider the diagra m R/ m ⊗ R W n ( A ) id ⊗ ¯ w i / / id ⊗ w 0   R/ m ⊗ R A/ m n +1 A ∼ r ⊗ a 7→ r a   A/ m A x 7→ x q i / / A/ m A. T o s how it commutes, it is enoug h to assume m is pr incipal, generated by π . Then commutativit y follows from the obvious congr uence w i ( a ) = a q i 0 + π a q i − 1 1 + . . . + π i a i ≡ a q i 0 mo d m A, for any element a = ( a 0 , a 1 , . . . ) π ∈ W ( A ). Therefore, id ⊗ ¯ w i is the comp os ition of a map whose k er nel is a nil ideal and a power of the F rob enius map. The scheme maps induced by both o f these are universal homeo mo rphisms. Now let us show that id ⊗ w 0 (whic h equals id ⊗ ¯ w 0 ) is a surjection with squar e - zero kernel. The ma p id ⊗ w 0 is s urjective by 1.21 (or 4.4). So let us s how the squa re of its kernel is zer o . By 4.4, the kernel of the map W n ( A ) → R / m ⊗ R A is the ideal V 1 W n ( A ) + m W n ( A ). Hence it is enough to show  V 1 W n ( A )  2 ⊆ m W n ( A ). This follows fro m (4.2.6).  THE BASIC GEOMETR Y OF W ITT VECTORS, I 29 6.9. Prop ositio n. L et ( B i ) i ∈ I b e a family of A -algebr as such t hat the induc e d map ` i Spec B i → Sp ec A is surje ctive. Then t he induc e d map a i Spec W R,E ,n ( B i ) → Sp ec W R,E ,n ( A ) is surje ctive. Pr o of. By 5.4, it is eno ugh to assume E consis ts of one ideal m . F urther, it is enough to show surjectivity after base change to R [1 / m ] and to R/ m . F or R [1 / m ], it follows from 6.1 and the equality W n ( C ) = C [0 ,n ] , when m is the unit ideal. Now consider base change to R/ m . By 6.8, the ring W n ( A ) / m W n ( A ) is a nilp otent extension o f A/ m A , and likewise for each B i , and so we are r educed to showing that ` i Spec B i / m B i → Spec A/ m A is surjective. This is true s ince base change distributes over disjoint unio ns and preserves surjectivity .  6.10. Prop ositi on. The R - algebr a Λ R,E ,n is ﬁnitely pr esen t e d, a nd the functor W R,E ,n pr eserves ﬁlter e d c olimits of R - algebr as. Pr o of. Since W R,E ,n is repr e sented by Λ R,E ,n , the t wo statements to b e proved are equiv alent. By 5.4, we may assume E consis ts of a s ingle ideal m . By EGA IV 2.7.1 [13], the ﬁrst statement can be veriﬁed fpqc lo ca lly on R , and in particular after ba se change to R [1 / m ] and to R m . The r efore b y (2.6.1), we can assume m is generated by a single element π . B ut b y 3 .3, the R -algebr a Λ R,E ,n is genera ted by the ﬁnite set θ π , 0 , . . . , θ π ,n .  7. Some general descent The pur po se of this section is to reco rd some facts a b out des cent of ´ etale algebr as which we will use to prov e the main theorem (9.2). The results men tion nothing ab out Witt vectors or anything else in this pa pe r . So it would be reas o nable to skip this section and r efer back to it only as needed. More precisely , we do the following. First, we set up some langua ge and notation for descent. This is essentially a rep etition of par ts of Grothendieck’s TDTE I [15]. (It could no t b e anything but.) Seco nd, w e prove an abstra ct r esult (7.10) rela ting gluing data a nd des c ent data for certain simple gluing co nstructions. Third, we recall Gr othendieck’s theorem (7.11) o n integral descent of ´ etale maps. Finally , w e prov e 7.1 2, which provides the plan of the pro of of 9.2. Aside from the languag e o f descent, only these three results will b e used outside this section. L anguage 7.1. Fib er e d c ate gories. Le t C b e a categor y with ﬁb ered pro ducts. Let E b e a category ﬁb ered ov er C . (See [15], A.1.1, or [1], VI.6.1.) F or any ob ject S o f C , let E S denote the ﬁb er of E o ver S . L e t us say tha t a map q : T → S in C is an E -equiv alence if q ∗ : E S → E T is an equiv alence o f categ ories, and let us say that q is a u niversal E -e quivalenc e if for any map S ′ → S in C , the base change q ′ : S ′ × S T → S ′ is an E - equiv alence. F o r the applications in the next section, the reader can tak e C = the categor y o f aﬃne sc hemes , E = the ﬁber e d categ ory ov e r C where E S is the category of a ﬃne ´ etale S -schemes and the functors q ∗ are given by bas e change. (7.1.1) Then a ny c losed immers ion deﬁned by a nil idea l is a universal E -equiv alence (EGA IV 18 .1 .2 [14]). 30 J. BOR GE R 7.2. Comp osition notation. Let S be an ob ject of C , and let C S × S denote the category of ob jects over S × S . Tha t is, a n o b ject of C S × S is a pair ( T , π T ), where T is an ob ject o f C and π T is a map T → S × S , ca lled its structur e map; a morphism is a morphism in C commuting with the maps to S × S . F or such an ob ject, let π T , 1 , π T , 2 denote the comp ositio n of the structure map T → S × S with the pro jections pr 1 , pr 2 : S × S → S . ( π T , 1 is the ‘source,’ and π T , 2 is the ‘target’.) W e will often abusiv ely leav e π T implicit and s ay that T is a n ob ject of C . Let 1 S denote the ob ject ( S, ∆) of C S × S , where ∆ : S → S × S is the diag onal map. Given tw o ob jects T , U ∈ C S × S , deﬁne T U ∈ C S × S as follows. As a n ob ject of C , it is the ﬁb ered pro duct (7.2.1) T U pr 1 / / pr 2   T π T , 2   U π T , 1 / / S. W e give T U the struc tur e of an ob ject of C S × S with the map (7.2.2) T U = T × S U ( π T , 1 ◦ pr 1 ,π U, 2 ◦ pr 2 ) / / S × S. 7.3. Cate gory obje cts and e quivalenc e r elations. A catego ry ob ject ov er S is an ob ject R ∈ C S × S together with maps (7.3.1) e R : 1 S → R, c R : R R → R in C S × S (called identit y and comp ositio n) satisfying the usua l identit y a nd asso cia - tivit y a xioms in the deﬁnition of a categor y . A mo rphism f : R → R ′ of suc h category ob jects deﬁned to b e a morphism in C S × S satisfying the functor axioms, that is , such that f ◦ e R = e R ◦ f and c R ′ ◦ f f = f ◦ c R , where f f denotes the map RR → R ′ R ′ induced by f . A categor y-ob ject structure on a sub ob ject R ⊆ S × S is a prop erty of R in that when it exis ts , it is unique . One might say that R is a reﬂexive transitive r e lation on S . W e say R is a n equiv ale nce relation on S if, in addition, the endo morphism (pr 2 , pr 1 ) of S × S tha t switches the tw o factors restricts to a map s : R → R (whic h is of course unique when it exists). 7.4. Pr e-actions (gluing data). Let T b e a n ob ject o f C S × S . A pr e-action of T on an ob ject X ∈ E S is deﬁned to b e a n isomor phis m (7.4.1) ϕ : π ∗ T , 2 ( X ) ∼ − → π ∗ T , 1 ( X ) in E T . A pr e-action is also ca lled a gluing datum o n X relative to the pair of maps ( π T , 1 , π T , 2 ). (Actually , Grothendiec k [1 5], A.1.4, calls ϕ − 1 the gluing datum.) Let PreAct( T , X ) denote the set of pr e -actions o f T on X . Any ma p T → T ′ in C S × S naturally induces a map PreAct( T ′ , X ) − → P reAct( T , X ) . THE BASIC GEOMETR Y OF W ITT VECTORS, I 31 If f : X → X ′ is a morphis m in E S betw een ob jects X , X ′ with pre- actions ϕ, ϕ ′ , then we s ay f is T -e quivariant if the fo llowing diagr a m co mmu tes: π ∗ T , 2 ( X ) π ∗ T , 2 ( f ) / / ϕ   π ∗ T , 2 ( X ′ ) ϕ ′   π ∗ T , 1 ( X ) π ∗ T , 1 ( f ) / / π ∗ T , 1 ( X ′ ) . In this w ay , the ob jects of E S equipp e d with a pre- action of T form a category . 7.5. A ctions. Now let R b e a categ ory ob ject ov er S . An action of R o n X is deﬁned to b e a pr e-action ϕ of R on X such that the diagr am e ∗ π ∗ R, 2 ( X ) e ∗ ( ϕ ) / / ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ e ∗ π ∗ R, 1 ( X ) ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ id ∗ S ( X ) and the dia gram c ∗ π ∗ R, 2 ( X ) c ∗ ( ϕ ) / / q q q q q q q q q q q q q q q q q q q q q q q q q q q q c ∗ π ∗ R, 1 ( X ) pr ∗ 2 π ∗ R, 2 ( X ) pr ∗ 2 ( ϕ ) & & ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ pr ∗ 1 π ∗ R, 1 ( X ) ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ pr ∗ 2 π ∗ R, 1 ( X ) pr ∗ 1 π ∗ R, 2 ( X ) , pr ∗ 1 ( ϕ ) 8 8 q q q q q q q q q q q q q q commute. Here, pr 1 and pr 2 denote the pro jections RR → R onto the ﬁrst and sec- ond factors, and the mo r phisms re pr esented b y eq uality signs are the isomor phis ms induced by the cano nical str uctur e maps ( g ◦ f ) ∗ ∼ → f ∗ ◦ g ∗ (notated c f ,g in [1 5], A.1.1(ii)) of the ﬁbe r ed ca tegory E corresp onding to the equalities π R, 2 ◦ e = id S = π R, 1 ◦ e and π R, 2 ◦ c = π R, 2 ◦ pr 2 , π R, 1 ◦ pr 2 = π R, 2 ◦ pr 1 , π R, 1 ◦ c = π R, 1 ◦ pr 1 . W e will often use the following more succinct, if slightly abusive, express ions of the commutativit y of the diagr ams a b ove: (7.5.1) e ∗ ( ϕ ) = id X , c ∗ ( ϕ ) = (pr ∗ 1 ϕ ) ◦ (pr ∗ 2 ϕ ) . Let Act( R , X ) denote the set of actions of R o n X . A morphism R → R ′ of category o b jects induces a map Act( R ′ , X ) − → Act( R, X ) in the ob vious wa y . 32 J. BOR GE R Last, note that if R is an eq uiv a lence r elation, then the diagram s ∗ π ∗ R, 2 ( X ) s ∗ ( ϕ ) / / s ∗ π ∗ R, 1 ( X ) π ∗ R, 1 ( X ) ϕ − 1 / / π ∗ R, 2 . commutes. This follows immediately from (7.5.1). The abbrevia ted version is (7.5.2) s ∗ ( ϕ ) = ϕ − 1 . 7.6. Desc ent data. Let q : S ′ → S be a ma p in C , and put R ( S ′ /S ) = S ′ × S S ′ . View R ( S ′ /S ) as an ob ject in C S ′ × S ′ by taking π R ( S ′ /S ) to b e the evident monomo r- phism R ( S ′ /S ) = S ′ × S S ′ − → S ′ × S ′ Then R ( S ′ /S ) is a n equiv alence rela tion on S ′ . An a c tion ϕ of R ( S ′ /S ) on a n ob ject X ′ of E S ′ is also c a lled a descent datum on X ′ from S ′ to S . (Aga in, it is actually ϕ − 1 that is called the descent datum in [1 5].) W e might call R ( S ′ /S ) the descent, or Galois, gro upo id o f the map q : S ′ → S . Because the tw o comp ositions R ( S ′ /S ) = S ′ × S S ′ ⇒ S ′ → S are equa l, for any ob ject X ∈ E S , the ob ject q ∗ ( X ) o f E S ′ has a canonical pre -action of R ( S ′ /S ), a nd it is eas y to check that this is an ac tion. W e say that q is a desc ent map for the ﬁber ed catego ry E if the functor from E S to the ca tegory of ob jects of E S ′ with an R -actio n is fully faithful. W e say it is an eﬀe ctive desc ent map if it is an equiv alence. 7.7. When gluing data is desc ent data. Now supp ose we hav e a diagram (7.7.1) S ′′ / / / / S ′ / / S in C such that the t wo co mpo sitions S ′′ ⇒ S are equal. The universal pro p er ty o f pro ducts gives a map S ′′ − → S ′ × S S ′ = R ( S ′ /S ) . F o r any ob ject X ′ ∈ E S ′ , this map induces a function Act( R ( S ′ /S ) , X ′ ) − → PreAct( S ′′ , X ′ ) . Let us say that gluing data on X ′ is desc ent data relative to the diag ram (7.7.1) when this map is a bijection. Gluing two obje cts Here we sp ell out in p erhaps excessive detail some basic fac ts ab o ut equiv ale nce relations on disjoin t unions which are E -tr ivial (though not necessarily trivia l) o n each factor . F r om now on, le t C denote the category of aﬃne schemes, s chemes, or a lg ebraic spaces. (W e o nly need so me w ea k h yp o theses o n copro ducts in C , but let us not bo ther to determine which o nes we nee d.) THE BASIC GEOMETR Y OF W ITT VECTORS, I 33 7.8. Equivalenc e r elations on a disjoint union. Supp ose S is a copro duct S a + S b of tw o ob jects S a , S b ∈ C . (W e us e the symbols a, b to index the summands only to emphas iz e their distinctio n from the symbols 1 , 2 that index the factors in the pro duct S × S .) Le t R be an equiv alenc e r elation on S , and let R ij denote R × S × S ( S i × S j ), for an y i, j ∈ { a , b } . Let π R ij , 1 denote the e vident co mpo sition R ij = R × S × S ( S i × S j ) pr 1 − → S i and π R ij , 2 the ana lo gous map R ij → S j . W e will so metimes view R ij as an o b ject of C S × S using the induced map R ij → S i × S j → S × S . Let e i : S i → R ii and c ij k : R ij R j k → R ik and s ij : R ij → R j i denote the evident restrictions of e and c a nd s . 7.9. A ctions over a disjoint union. F o r a ny ob ject X over S , write X a = S a × S X and X b = S b × S X . F o r any pre -action (7.9.1) ϕ : π ∗ R, 2 X − → π ∗ R, 1 X , of R o n X , let us write ϕ ij for the res tr iction of ϕ to R ij . In order for this pre-actio n to b e a n ac tion, it is necessary and suﬃcien t that for all i, j, k ∈ { a , b } we hav e e ∗ i ( ϕ ii ) = id X i and (7.9.2) c ∗ ij k ( ϕ ik ) = pr ∗ 1 ( ϕ ij ) ◦ pr ∗ 2 ( ϕ j k ) . (7.9.3) This is just a restatement of (7.5.1), summand by summand. In that case, 7.5.2 bec omes (7.9.4) s ∗ ij ( ϕ j i ) = ϕ − 1 ij . 7.10. Prop osition. L et R b e an e qu ivalenc e r elation on S = S a + S b such that for i = a, b , the map e i : S i → R ii is a universal E -e quivalenc e. Then for any obje ct X ∈ E S , the map Act( R, X ) ϕ 7→ ϕ ba / / PreAct( R ba , X ) is a bije ction. Pr o of. Let us ﬁrst show injectivity . Let ϕ and ϕ ′ be actions of R o n X such that ϕ ba = ϕ ′ ba . W e nee d to show that this implies ϕ ij = ϕ ′ ij for a ll i , j ∈ { a, b } . Consider each case sepa rately . F or i j = b a , it is true by as sumption. When ij = ab , equation (7.9 .4) and the given equality ϕ ba = ϕ ′ ba , imply ϕ ab = s ∗ ba ( ϕ ab ) − 1 = s ∗ ba ( ϕ ′ ab ) − 1 = ϕ ′ ab . When i = j , since e i is an E - equiv alence, it is enoug h to show e ∗ i ( ϕ ii ) = e ∗ i ( ϕ ′ ii ). But by (7.9.2), we hav e e ∗ i ( ϕ ii ) = id X i = e ∗ i ( ϕ ′ ii ) . Therefore ϕ = ϕ ′ , which prov es injectivity . Now consider s ur jectivity . Let ϕ ba be a pre- a ction of R ba on X . Deﬁne (7.10.1) ϕ ab = s ∗ ab ( ϕ ba ) − 1 and for i = a, b deﬁne ϕ ii to b e the map such that (7.10.2) e ∗ i ( ϕ ii ) = id X i , which exists and is unique b ecause e i is an E -equiv a lence. W e need to chec k that the pre-action ϕ = ϕ aa + ϕ ab + ϕ ba + ϕ bb of R on X is actually an a ction. T o do this, we will verify the r elations (7.9.2) and (7.9.3). The identit y axiom (7 .9 .2) holds becaus e it is the deﬁning pr op erty (7.10.2) of ϕ ii . 34 J. BOR GE R Now consider the asso ciativity a x iom (7.9.3) for the v ar ious po ssibilities for ij k . Since i, j, k ∈ { a , b } , tw o of i , j, k must b e equal. If i = j , the compo sition f R j k pr − 1 2 ∼ / / S j j R j k e j × id / / R j j R j k is an E -equiv alence, b ecause it is a base c ha ng e o f the universal E -equiv a lence e j . Therefore it is enough to show (7.10.3) f ∗ c ∗ j j k ( ϕ j k ) = f ∗ pr ∗ 1 ( ϕ j j ) ◦ f ∗ pr ∗ 2 ( ϕ j k ) . By the equalit y pr 1 ◦ f = e j ◦ π R jk , 1 and (7.10.2), w e ha ve f ∗ pr ∗ 1 ( ϕ j j ) = π ∗ R jk , 1 e ∗ j ( ϕ j j ) = π ∗ R jk , 1 (id X j ) = id . On the other hand, by c j j k ◦ f = id R jk = pr 2 ◦ f , we ha ve f ∗ c ∗ j j k ( ϕ j k ) = f ∗ pr ∗ 2 ( ϕ j k ). Equation (7 .1 0.3) then follows. The case j = k is similar to the case i = j . (O r apply s to the case i = j .) Last, supp ose i = k . The following diagra m is easily check ed to b e c artesian: (7.10.4) R ij (id R ij ,s ij ) / / π R ij , 1   R ij R j i c iji   S i e i / / R ii . (This is just another expre s sion of the exis tence and uniqueness of inv erse s in a group oid.) Since e i is a universal E -eq uiv a lence, (id R ij , s ij ) is an E -eq uiv a lence. So it is eno ugh to show axio m (7.9.3) after applying (id R ij , s ij ) ∗ , that is, to s how (7.10.5) (id R ij , s ij ) ∗ c ∗ ij i ( ϕ ii ) = (id R ij , s ij ) ∗ pr ∗ 1 ( ϕ ij ) ◦ (id R ij , s ij ) ∗ pr ∗ 2 ( ϕ j i ) . By the comm utativity of (7 .10.4) and (7.10.2), we hav e (id R ij , s ij ) ∗ c ∗ ij i ( ϕ ii ) = π ∗ R ij , 1 e ∗ i ( ϕ ii ) = π ∗ R ij , 1 (id X i ) = id . Combining this with the e q uation ϕ j i = s ∗ j i ( ϕ ij ) − 1 , equa tion (7.10.5) re duces to (id R ij , s ij ) ∗ pr ∗ 1 ( ϕ ij ) = (id R ij , s ij ) ∗ pr ∗ 2 s ∗ j i ( ϕ ij ) . But this holds b ecause we hav e pr 1 ◦ (id R ij , s ij ) = id R ij = s j i ◦ s ij = s j i ◦ pr 2 ◦ (id R ij , s ij ) . Therefore the equations in (7.9.3) hold for all i , j, k , a nd so the pr e-action is an action.  Gr othendie ck’ s the or em Recall that a map Spec B → Spec A o f aﬃne schemes is said to b e integral if the corres p o nding r ing map A → B is integral (and no t necessar ily injectiv e). 7.11. Theorem . Every surje ctive int e gr al map Y → X of aﬃne schemes is an eﬀe ctive desc ent map for the ﬁb er e d c ate gory E over C of (7.1.1). This theo rem is prov en in SGA 1 IX 4.7 [1] up to t wo details. First, the arg umen t given there cov er s only morphisms Y → X w hich are ﬁnite a nd of ﬁnite pr e sentation; and second, the statement there has no aﬃneness in the ass umptions o r in the conclusion. The ﬁrst point ca n b e handled b y a standard limiting a rgument (or one can apply Theorem 5.17 plus Remark 2.5 (1b) in Rydh [22]). The second p oint can b e handled w ith Chev alley ’s theorem; the form most co nv enient here would the THE BASIC GEOMETR Y OF W ITT VECTORS, I 35 ﬁnal one, Rydh’s [21] Theor em (8.1), which is free of no etheria nness, sepa ratedness, ﬁniteness, a nd scheme-theoretic a ssumptions. Gluing and desc ent of ´ etale algebr as 7.12. Prop osition. Consider a diagr am of rings (7.12.1) B d / / B ′ h 1 / / h 2 / / B ′′ A f / / e O O A ′ g 1 / / g 2 / / e ′ O O A ′′ e ′′ O O such that h i ◦ e ′ = e ′′ ◦ g i , for i = 1 , 2 . Al so assu m e the fol lowing pr op erties ar e satisﬁe d: (a) the two p ar al lel right-hand squar es ar e c o c artesian, (b) b oth r ows ar e e qualizer dia gr ams, (c) r elative to t he lower r ow, gluing data on any ´ etale A ′ -algebr a is desc ent data, (d) f satisﬁes eﬀe ct ive desc ent for the ﬁb er e d c ate gory of ´ etale algebr as, and (e) e ′ is ´ etale. Then e is ´ etale and the left-hand s qu ar e is c o c artesian. Note that when w e use the language of des cent in the category of rings (as in (c) and (d)), we understand that it refers to the corr esp onding statements in the opp osite categ ory . Pr o of. Pr op erty (a) equips the ´ etale A ′ -algebra B ′ with g luing da ta ϕ relative to ( g 1 , g 2 ). Indeed, tak e ϕ to b e the compo sition A ′′ ⊗ g 1 ,A ′ B ′ ∼ − → B ′′ ∼ − → A ′′ ⊗ g 2 ,A ′ B ′ . By prop er ty (c), this gluing data comes from unique des cent data r elative to f . Therefore by (d) and (e), the A ′ -algebra B ′ descends to a n ´ etale A -alg ebra C . Now apply the functor C ⊗ A − to the low er row o f diagra m (7.12.1). By (a) and the deﬁnition of descent, the res ult can be identiﬁ ed with the s e quence C / / B ′ h 1 / / h 2 / / B ′′ . This se q uence is als o an eq ua lizer diag ram, b ecause the low er r ow of (7 .12.1) is an equalizer diagram, by (b), a nd b eca use C is ´ etale ov er A and hence ﬂat. Again by (b), the upp er row of (7.12 .1) is an equalizer diagram, and so we hav e C = B . Therefore, B is a n ´ etale A -algebra a nd the left-hand square is co cartesian.  8. Ghost descent in the single-prime case W e r eturn to the notation of 1.2. Supp ose E cons is ts o f a single maximal ideal m , and ﬁx an integer n > 1. W rite W n = W R, m ,n , and so o n. Let A b e an R - a lgebra, and let α n denote the map (8.0.2) W n ( A ) α n − − → W n − 1 ( A ) × A given by the ca nonical pro jection on the factor W n − 1 ( A ) and the n -th ghost c o m- po nent w n on the fac tor A . Let I n ( A ) denote the kernel of α n . F or example, if m is genera ted by π , then in terms of the Witt co mpo nents, we have (8.0.3) I n ( A ) = { (0 , . . . , 0 , a ) π ∈ A [0 ,n ] | π n a = 0 } . 8.1. Prop ositio n. We have the fol lowing: 36 J. BOR GE R (a) α n is an inte gr al ring homomorphism. (b) The kernel I n ( A ) of α n is a squar e-zer o ide al. (c) If A is m -ﬂat, then α n is inje ct ive. (d) The diagr am W n − 1 ( A ) ¯ w n / / A/ m n A W n ( A ) w n / / O O O O A, O O O O wher e t he vertic al maps ar e the c anonic al ones, is c o c artesian. (e) View A as a W n ( A ) -algebr a by t he map w n : W n ( A ) → A . Then every element of the kernel of the multiplic ation map A ⊗ W n ( A ) A − → A is nilp otent . (f ) In the diagr am (8.1.1) W n ( A ) α n / / W n − 1 ( A ) × A ¯ w n ◦ pr 1 / / pr 2 / / A/ m n A, wher e pr 2 denotes the r e duction of pr 2 mo dulo m n , t he image of α n agr e es with the e qualizer of ¯ w n ◦ pr 1 and pr 2 . Pr o of. (a): It is enough to show that ea ch factor o f W n − 1 ( A ) × A is in teg ral o ver W n ( A ). The ﬁrst factor is a q uotient ring , and hence in tegral. Now cons ide r an element a ∈ A . T hen a q n is the image in A of the T eichm¨ uller lift [ a ] ∈ W n ( A ). (See 1.2 1.) Therefor e the second factor is also integral ov er W n ( A ). (b): It suﬃces to show this after base change to R [1 / m ] × R m . Therefore, by 6.1, we may assume m is gene r ated b y a single element π . Then an element of the kernel of α n will b e of the for m V n π [ a ] = (0 , . . . , 0 , a ) π , where π n a = 0. On the other hand, by (4.2.6) w e ha ve ( V n π [ a ])( V n π [ b ]) = π n V n π [ ab ] = (0 , . . . , 0 , π n ab ) π = 0 . (c): W e hav e ( w 6 n − 1 × id A ) ◦ α n = w 6 n . Since A is m -ﬂat, the map w 6 n is injectiv e (2.7), and hence s o is α n . (d): As above, it is e no ugh by 6.1 to assume m is generated b y a single ele men t π . Then we hav e A ⊗ W n ( A ) W n − 1 ( A ) = A ⊗ W n ( A ) W n ( A ) /V n W n ( A ) = A/w n ( V n W n ( A )) A. Examining the Witt p olynomials (3.1.1) shows w n ( V n W n ( A )) = π n A . (e): Aga in, by 6.1 we ma y ass ume m is g enerated by a single ele ment π . T o show every element x ∈ I is nilp otent, it is enough to r estrict x to a set of generator s . Therefore it is enough to show (1 ⊗ a − a ⊗ 1 ) q n = 0 for every element a ∈ A . Now supp ose that, for j = 0 , . . . , q n , we could show (8.1.2)  q n j  a j ∈ im( w n ) . Then we would hav e (1 ⊗ a − a ⊗ 1) q n = X j ( − 1) j  q n j  a j ⊗ a q n − j = X j ( − 1) j ⊗  q n j  a j a q n − j = (1 ⊗ a − 1 ⊗ a ) q n = 0 , which would complete the pro of. So let us show (8.1.2). THE BASIC GEOMETR Y OF W ITT VECTORS, I 37 Let f = or d p ( q ) a nd i = ord p ( j ). Then we hav e ord p  q n j  = or d p ( q n j − 1 ) + ord p  q n − 1 j − 1  > nf − i. It follows tha t  q n j  a j is an R -linear m ultiple of π nf − i a j . Since w n is an R -algebr a map, it is therefore enough to show (8.1.3) π nf − i a j ∈ im( w n ) . Now, for b ∈ A a nd s = 0 , . . . , n , we hav e π n − s b q s = w n ( V n − s π [ b ]), and therefore π n − s b q s is in the imag e of w n . So to s how (8.1.3), it is enough to ﬁnd an integer s and an element b ∈ A such that π n − s b q s is an R -linear diviso r of π nf − i a j . In particular, it is suﬃcient for b and s to satisfy b q s = a j and n − s 6 nf − i . T ake s to be the greatest integer at most if − 1 . Then we hav e q s | j ; so if w e set b = a j /q s ∈ A , we hav e b q s = a j . It remains to show n − s 6 nf − i . This is equiv alent to n − if − 1 6 nf − i , which is in tur n equiv a lent to (1 − f )( n − if − 1 ) 6 0 . And this holds becaus e 1 − f 6 0 and n − if − 1 > 0. (Recall that j 6 q n .) This completes the pro of of (e). (f): As ab ov e, w e may assume that m ca n be gener ated by a single element π . F o r any element a = ( a 0 , . . . , a n ) π ∈ W n ( A ), we hav e α n ( a ) =  ( a 0 , . . . , a n − 1 ) , a q n 0 + · · · + π n − 1 a q n − 1 + π n a n  . Therefore an element  ( a 0 , . . . , a n − 1 ) , b  ∈ W n − 1 ( A ) × A lies in the imag e of α n if and only if a q n 0 + · · · + π n − 1 a q n − 1 ≡ b mo d m n A, which is exactly what we needed to show.  8.2. Corollary . F or any R -algebr a A , the ghost m ap w 6 n : W n ( A ) − → A [0 ,n ] is inte gr al, and its kernel J satisﬁes J 2 n = 0 . Pr o of. By 8.1 and induction on n .  8.3. Theorem. (a) The map α n is a n eﬀe ctive desc ent map for the ﬁb er e d c ate gory of ´ etale algebr as. (b) R elative to the diagr am (8.3.1) W n ( A ) α n / / W n − 1 ( A ) × A ¯ w n ◦ pr 1 / / pr 2 / / A/ m n A, gluing data on any ´ etale W n − 1 ( A ) × A -algebr a is desc ent data (7.7). (c) If A is m -ﬂat, then for any A ′ -algebr a B ′ e quipp e d with gluing data ϕ , the desc ende d A -algebr a is the subring B of B ′ on which t he fol lowing diagr am c ommutes: A/ m n A ⊗ ¯ w n ◦ pr 1 B ′ B ′ 1 ⊗ id B ′ 6 6 ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ 1 ⊗ id B ′ ( ( P P P P P P P P P P P P A/ m n A ⊗ pr 2 B ′ . ϕ O O 38 J. BOR GE R Pr o of. (a): This follo ws fr om Gr othendieck’s theorem 7 .11 and 8.1(a)–(b). (b): W e will use 7.1 0, wher e C and E are as in (7.1.1). In the nota tion of 7.8, put S a = Sp ec W n ( A ) and S b = Sp e c A . Let Γ b e the equiv alence r elation S × Sp ec W n ( A ) S on S . By 8 .1(d), we have Γ ba = Sp ec A/ m n A . The map e a is an is o morphism bec ause W n − 1 ( A ) is a quo tient r ing of W n ( A ). T he map e b is a nil immersion, by 8.1(e), and hence is an E - equiv alence. Thus w e ca n apply 7.1 0, which s ays that a Γ-action is the same as a Γ ba pre-action. In other words, gluing data is descen t data. (c): This will follow from 7 .12 once we v er ify the h yp otheses. 7.1 2(a)–(b) are clear; 7.12(c) follows from (b) ab ov e ; 7.1 2(d) follows from (a) a bove; and 7.12(e) follows from the deﬁnition o f B , for the top row o f (7 .1 2.1), and from 8.1(c) a nd (f), for the b ottom row.  8.4. R emark. F or a ny ring C , let EtAlg C denote the catego ry of ´ etale C -a lgebras. Then another way of expressing par t (b) of this theorem is that the induced functor EtAlg W n ( A ) − → EtAlg W n − 1 ( A ) × EtAlg A/ m n A EtAlg A is an equiv alence. (Of cours e, the ﬁbe red pro duct of categ ories is taken in the weak sense.) In par ticular, we can prov e things a b out ´ etale W n ( A )-algebras by induction on n . This is the main technique in the pro o f of 9.2. But it a lso see ms interesting in its own right and will pro bably hav e a pplica tions b e yond the present pap er. 8.5. R emark. If w e let ¯ W n ( A ) deno te the imag e of α n , then the induced dia gram ¯ W n ( A ) / / W n − 1 ( A ) × A ¯ w n ◦ pr 1 / / pr 2 / / A/ m n A. satisﬁes a ll the conclusions of the theore m ab ove, reg ardless of whether A is m -ﬂat. Indeed, it is an equalizer diagram by 8.1(f) and the deﬁnition of ¯ W n ( A ); it is an eﬀective descent map by 8.3 and 8 .1(b). Last, beca use ¯ W n ( A ) is the ima g e o f α n , gluing (resp. desce nt ) data relative to ¯ W n ( A ) agrees with gluing (resp. descent) data rela tive to W n ( A ). In par ticular, gluing data rela tive to ¯ W n ( A ) is descent data. 9. W and ´ et ale morphisms W e return to the gener al context of 1.2. In pa rticular, E is no lo nger req uired to consist o f one ideal. 9.1. Lemma. Consider a c ommutative squar e of aﬃne schemes (or any schemes) X f   X ′ f ′   g o o Y Y ′ , h o o and let U b e an op en subscheme of Y . Supp ose the fol lowing hold (a) f and f ′ ar e ´ etale, (b) the squar e ab ove b e c omes c artesian after t he b ase change U × Y − , (c) g and h b e c ome surje ctive and universal ly inje ctive after the b ase change ( Y − U ) × Y − . Then the squar e ab ove is c artesian. Pr o of. Let e denote the induced map ( g , f ′ ) : X ′ → X × Y Y ′ . It is enoug h to show e is ´ etale, s urjective, a nd universally injective (EGA IV (17 .9.1) [14]). The comp osition of e with pr 2 : X × Y Y ′ → Y ′ is f ′ . Because f is ´ etale, so is its base THE BASIC GEOMETR Y OF W ITT VECTORS, I 39 change pr 2 . Combinin g this with the ´ etaleness of f ′ implies that e is ´ etale (E GA IV 17 .3 .4 [14]). The surjectivit y and univ er sal injectivit y o f e ca n be check ed after ba se ch ange ov er Y to U and to Y − U . By assumption e becomes an isomorphism after base change to U . In par ticula r, it becomes surjective and universally injective. Let ¯ e, ¯ g, ¯ h denote the maps e, g , h pulled back fro m Y to Y − U . Let ¯ h ′ denote the ba se c hange of ¯ h from Y to X . Then, as above, we hav e ¯ g = ¯ h ′ ◦ ¯ e . Since ¯ h is universally injectiv e , so is ¯ h ′ . Combining this with the fact that ¯ g is universally injectiv e, implies that ¯ e is as well (EGA I 3.5.6 –7 [12]). Finally ¯ e is surjective since ¯ h ′ is injective and ¯ g is s urjective.  9.2. Theorem. F or any ´ etale map ϕ : A → B and any element n ∈ N ( E ) , the induc e d map W R,E ,n ( ϕ ) : W R,E ,n ( A ) → W R,E ,n ( B ) is ´ et ale. Pr o of. By 5.4, it is eno ugh to assume E cons ists of a s ing le maximal ideal m . Also, it will simplify notation if we ass ume m is principal, generated b y an element π . W e may do this by by 6 .1 and b ecaus e it is enough to show ´ etaleness after applying R m ⊗ R − and R [1 / m ] ⊗ R − . Let us write W n = W R,E ,n . W e will r eason by induction on n , the case n = 0 b eing clear beca us e W 0 is the ident it y functor . So from now on, a ssume n > 1. Let ¯ W n ( A ) deno te the image of α n : W n ( A ) → W n − 1 ( A ) × A , and le t ¯ α n denote the induced injection ¯ W n ( A ) → W n − 1 ( A ) × A . Deﬁne ¯ W n ( B ) and ¯ α n for B similar ly . Step 1 : ¯ W n ( B ) is ´ etale over ¯ W n ( A ) . T o do this, it suﬃces to verify conditions (a)–(e) of 7.1 2 for the following diag ram ¯ W n ( B ) ¯ α n / / W n − 1 ( B ) × B ¯ w n ◦ pr 1 / / pr 2 / / B / m n B ¯ W n ( A ) ¯ α n / / O O W n − 1 ( A ) × A ¯ w n ◦ pr 1 / / pr 2 / / O O A/ m n A, O O where the vertical maps ar e induced by ϕ and functoriality . W e know 7.12(a) holds by induction. Conditions 7.12(c)–(d) hold by 8.3 (or 8 .5). Condition 7.12(e) w as shown alrea dy in 8.1(f). Now consider 7.12(b). It is clea r that the square of pr 2 maps is co car tesian. So, all tha t remains is to check that the squa r e of ¯ w n ◦ pr 1 maps is co cartesian. By induction, W n − 1 ( B ) is ´ etale o ver W n − 1 ( A ), and so this follows fro m 9.1, which we can apply by 6 .1 and 6 .8. Step 2 : W n ( B ) is ´ etale over W n ( A ). By 8 .1(b), the kernel I n ( A ) of the map α n : W n ( A ) → ¯ W n ( A ) has squa r e zer o. Therefore by EGA IV 18.1 .2 [14], there is an ´ etale W n ( A )-algebra C and an isomor - phism f : C ⊗ W n ( A ) ¯ W n ( A ) → ¯ W n ( B ). Now consider the square C / / d % % ❏ ❏ ❏ ❏ ❏ ❏ ¯ W n ( B ) W n ( A ) O O / / W n ( B ) , O O O O where the upper map is the one induced by f and where d will so o n b e deﬁned. By 8.1(b), the kernel I n ( B ) of the rig ht -hand map has square zero . There fo re since C is ´ etale ov er W n ( A ), ther e exists a unique map d making the diagr am commute. Let us now show that d is an isomo r phism. 40 J. BOR GE R Because C is ´ etale and hence ﬂat ov er W n ( A ), we have a co mm utative diagr am with exac t rows: 0 / / I n ( B ) / / W n ( B ) / / ¯ W n ( B ) / / 0 0 / / C ⊗ W n ( A ) I n ( A ) / / e O O C / / d O O C ⊗ W n ( A ) ¯ W n ( A ) / / ∼ f O O 0 . So to show d is a n isomor phism, it is enough to s how e is an iso morphism. Because I n ( A ) is a square-ze r o ideal, the action of W n ( A ) on it factors through ¯ W n ( A ). Therefore, e factors as follows: C ⊗ W n ( A ) I n ( A ) = C ⊗ W n ( A ) ¯ W n ( A ) ⊗ ¯ W n ( A ) I n ( A ) f ⊗ id − → ¯ W n ( B ) ⊗ ¯ W n ( A ) I n ( A ) g − → I n ( B ) , Since f is an isomorphism, it is enough to s how g is an isomo rphism. Using the descriptio n (8.0.3) of I n , the map g can b e extended to the following commutativ e diag ram with exa ct rows: 0 / / I n ( B ) / / B · π n / / π n B / / 0 0 / / ¯ W n ( B ) ⊗ I n ( A ) / / g O O ¯ W n ( B ) ⊗ A · π n / / (pr 2 ◦ ¯ α n ) · ϕ O O ¯ W n ( B ) ⊗ π n A, (pr 2 ◦ ¯ α n ) · ϕ O O / / 0 where ⊗ denotes ⊗ ¯ W n ( A ) , for short. Therefore it is enough to sho w the rig ht t wo vertical maps ar e isomo rphisms, and to do this, it is enough to show the right-hand square in the diagra m W n ( B ) / / / / ¯ W n ( B ) pr 2 ◦ ¯ α n / / B W n ( A ) / / / / O O ¯ W n ( A ) O O pr 2 ◦ ¯ α n / / A O O is co car tesian. W e will do this b y applying 9.1, with U = Spec R [1 / m ] ⊗ R ¯ W n ( A ). By step 1, co nditio n 9.1(a) holds. No w consider conditions 9.1(b)–(c). By 8.3(b), the horizontal maps in the left-hand square hav e squar e - zero kernel. In particular , the scheme maps they induce ar e universal ho meomorphisms. And by 6.1, they bec ome isomor phisms after applying R [1 / m ] ⊗ R − . Therefor e it is enough to show 9.1(b)–(c) hold for the p erimeter of the diag ram a b ove. In this cas e, 9 .1(b) follows from 6.1, and 9.1(c) fo llows from 6.8.  9.3. R emark. O bserve that when A is E -ﬂat, the pro of terminates after step 1, which is just a n a pplica tion o f 7.12. Thus in the central case, the argument is not m uch more than 8.1 a nd some g e neral descent theory . 9.4. Corollary . L et B an ´ etale A -algebr a, and let C b e any A -algebr a. Then for any n ∈ N ( E ) , the induc e d diagr am W R,E ,n ( B ) / / W R,E ,n ( B ⊗ A C ) W R,E ,n ( A ) / / O O W R,E ,n ( C ) O O is c o c artesian. THE BASIC GEOMETR Y OF W ITT VECTORS, I 41 Pr o of. By 5.4, we can ass ume E consis ts of a single ideal m . The pro o f will b e completed by 9.1, once w e check its hypo theses ar e sa tisﬁed. Condition (a) of 9.1 holds by 9.2, c ondition (b) holds by 6.1 and 2.7, and c ondition (c) holds by 6 .8.  9.5. W n do es not gener al ly c ommut e with c opr o duct s . Almost a nythin g is an exam- ple. F or instance, with the p -typical Witt vectors, W 1 ( A ⊗ Z A ) is not iso morphic to W 1 ( A ) ⊗ W 1 ( Z ) W 1 ( A ), when A is F p [ x ] or Z [ x ]. 9.6. W do es not gener al ly pr eserve ´ etale maps. L e t W denote p -typical Witt func- tor (non-trunca ted), and let ϕ denote the evident inclusion Q [ x ] → Q [ x ± 1 ], whic h is ´ etale. While W ( ϕ ) is best viewed as a map of pro-ring s, it is p o ssible to view it as a map of ordinary rings, and ask whether it is ´ etale. It is not: W ( ϕ ) can be ident iﬁed with ϕ N : Q [ x ] N → Q [ x ± 1 ] N , whic h is not ´ etale b eca us e Q [ x ± 1 ] N is not ﬁnitely gene r ated as an Q [ x ] N -algebra . This is a n elementary exercise. 9.7. Other tru nc ation sets for the big Witt ve ctors. Some writer s hav e considered more general systems of truncations fo r the big Witt functor (1.15). See Hesselholt– Madsen [17], subsectio n 4.1, for example. Given a ﬁnite set T of p ositive int egers closed under extra ction of divisors , they deﬁne a n endo functor W T of the catego ry of r ings. When T consis ts of all the divisor s of s ome integer d > 1, then W T agrees with our W Z ,E ,n , where E consis ts o f the maximal ideals m ⊂ Z that co nt ain d and where n m = or d m ( d ). Thus the t wo systems of trunca tions are coﬁnal with res p e c t to each o ther. The functors W T also pres e r ve ´ etale maps. Indee d, it is enough to show that the base ch ange to Z [1 / T ] a nd to Z ( p ) , for each pr ime p ∈ T , is ´ etale. (See EGA IV (17.7.2 )(ii).) Applying the identit y W T ( A )[1 /p ] = W T ( A [1 /p ]), which c an b e established by lo oking at the g raded pie c e s of the V erschiebung ﬁltration, it is enough to consider Z [1 / T ]-a lg ebras and Z ( p ) -algebra s. In the either ca se, W T ( A ) is simply a pr o duct o f p -typical Witt ring s W n ( A ) fo r v ario us primes p and lengths n (see [17], (4.1 .10)), in whic h case the result follows from 9 .2, or v an der K allen’s original theor em [24], (2.4 ). References [1] R evˆ et ements ´ etales et gr oup e fondamental (SGA 1) . Do cuments Math´ ematiques (P aris ), 3. Soci´ et´ e M ath ´ ematique de F rance, P ar is, 2003. S ´ eminaire de g´ eom ´ etrie alg ´ ebrique du Bois Marie 1960–61. Di rected b y A. Grothendiec k, With tw o pap ers b y M. Ra ynaud, Up dated and annotated reprint of the 1971 ori ginal [Lecture Notes in Math., 224, Springer, Ber l in]. [2] F rancis Borceux. Handb o ok of c ate goric al algebr a. 1 , vo lume 50 of Encyclop ed i a of Mathe- matics and it s Applic ations . Cambridge Universit y Press, Cam br i dge, 1994. B asi c category theory . [3] F rancis Borceux. 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(3) , 20:619–643, 1970. [24] Wilberd v an der Kal len. D escen t f or the K -theory of p olynomial rings. Math. Z. , 191(3):405– 415, 1986. [25] Clarence Wilke rson. Lambda-rings, binomial domains, and v ector bundles ov er C P ( ∞ ). Comm. Algebr a , 10(3):311–328, 1982. [26] Ernst Witt. Zyklische K¨ or per und Algebren der Charakteristik p v om Grad p n . Struktur diskret bewertete r p erfekter K¨ orp er mit v oll ko mmenem R estklassen-k¨ orper der charakte ristik p . J. R eine Angew. Math. , (176), 1937. [27] Ernst Witt. Col le cte d p ap ers. Gesammelte Abhand lungen . Spri nger-V erlag, Berlin, 1998. With an essay by G ¨ unte r Harder on Witt ve ctors, Edi ted and with a preface in English and German b y Ina Kersten. Aust ralian Na tional Univ ersity E-mail addr ess : james.borg er@anu.e du.au

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